Integral Inequalities and Generalized Convexity 1032526327, 9781032526324

Table of contents :
Cover
Half Title
Title Page
Copyright Page
Contents
Foreword
Author Biographies
Preface
Symbol Description
1. Introduction
1.1. Generalized Convexity
1.2. Invexity
1.3. Integral Inequalities
1.4. Fractional Calculus
1.5. Majorization Inequalities
1.6. Time Scale Calculus
1.7. Interval Analysis
1.7.1. Interval arithmetic
1.7.2. Integral of interval-valued functions
1.8. Stochastic Processes
2. Integral Inequalities for Strongly Generalized Convex Functions
2.1. Introduction
2.2. Preliminaries
2.3. Hermite–Hadamard Type Inequalities for Functions Whose Derivatives are Strongly n-Convex
2.3.1. Application to special means
2.4. Weighted Version of Hermite–Hadamard Type Inequalities for Strongly GA-Convex Functions
2.5. Hermite–Hadamard Type Integral Inequalities for the Class of Strongly Convex Functions on Time Scales
3. Integral Inequalities for Strongly Generalized Convex Functions of Higher Order
3.1. Introduction
3.2. Preliminaries
3.3. Strongly Generalized Convex Functions of Higher Order
3.4. Integral Inequalities for Higher Order Strongly Exponentially Convex Functions
4. Integral Inequalities for Generalized Preinvex Functions
4.1. Introduction
4.2. Preliminaries
4.3. Hermite–Hadamard Type Inequalities via Preinvex Functions
4.4. Application to Special Means
4.5. Generalized (m,h)-Preunivex Mappings via k-Fractional Integrals
5. Some Majorization Integral Inequalities for Functions Defined on Rectangles via Strong Convexity
5.1. Introduction
5.2. Preliminaries
5.3. Majorization Integral Inequalities for Strong Convexity
6. Hermite–Hadamard Type Inclusions for Interval-Valued Generalized Preinvex Functions
6.1. Introduction
6.2. Preliminaries
6.3. Hermite–Hadamard Type Inclusions for Interval-Valued Preinvex Functions
6.4. Hermite–Hadamard Type Inclusions for Interval-Valued Coordinated Preinvex Functions
6.5. Hermite–Hadamard Type Fractional Inclusions for Harmonically h-Preinvex Interval-Valued Functions
7. Some Inequalities for Multidimensional General h-Harmonic Preinvex and Strongly Generalized Convex Stochastic Processes
7.1. Introduction
7.2. Preliminaries
7.3. General h−Harmonically Preinvex Stochastic Process (Gh − HPnφSP)
7.4. Multidimensional General h−Harmonic Preinvex Stochastic Processes (MGh − HPnφSP)
7.5. Strongly Generalized Convex Stochastic Processes
8. Applications
8.1. Hermite–Hadamard Inequality
8.1.1. Higher dimensional Hermite–Hadamard inequality
8.1.2. Mass transportation and higher dimensional Hermite–Hadamard inequality
8.2. Jensen’s Inequality
8.3. Time Scales
8.4. Interval-Valued Functions
8.4.1. Error estimation to quadrature rules Using Ostrowski type inequality for interval-valued functions
Bibliography
Index

Citation preview

Integral Inequalities and Generalized Convexity The book covers several new research findings in the area of generalized convexity and integral inequalities. Integral inequalities using various type of generalized convex functions are applicable in many branches of mathematics such as mathematical analysis, fractional calculus and discrete fractional calculus. The book contains integral inequalities of Hermite–Hadamard type, Hermite– Hadamard–Fejér type and majorization type for the generalized strongly convex functions. It presents Hermite–Hadamard type inequalities for functions defined on Time scales. Further, it provides the generalization and extensions of the concept of preinvexity for interval-valued functions and stochastic processes and gives Hermite–Hadamard type and Ostrowski type inequalities for these functions. These integral inequalities are utilized in numerous areas for the boundedness of generalized convex functions. Features:

•

• • •

Covers Interval-valued calculus, Time scale calculus, Stochastic processes – all in one single book. Numerous examples to validate results. Provides an overview of the current state of integral inequalities and convexity for a much wider audience, including practitioners. Applications of some special means of real numbers are also discussed.

The book is ideal for anyone teaching or attending courses in integral inequalities along with researchers in this area.

Integral Inequalities and Generalized Convexity

Shashi Kant Mishra Nidhi Sharma Jaya Bisht

First edition published 2024 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2024 Shashi Kant Mishra, Nidhi Sharma, Jaya Bisht Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright. com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. ISBN: 978-1-032-52632-4 (hbk) ISBN: 978-1-032-52765-9 (pbk) ISBN: 978-1-003-40828-4 (ebk) ISBN: 978-1-032-52767-3 (eBook+) DOI: 10.1201/9781003408284 Typeset in CMR10 font by KnowledgeWorks Global Ltd. Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.

Contents

Foreword

ix

Author Biographies

xi

Preface

xiii

Symbol Description

xvii

1 Introduction 1.1 1.2 1.3 1.4 1.5 1.6 1.7

1.8

Generalized Convexity . . . . . . . . . . . . Invexity . . . . . . . . . . . . . . . . . . . . Integral Inequalities . . . . . . . . . . . . . Fractional Calculus . . . . . . . . . . . . . Majorization Inequalities . . . . . . . . . . Time Scale Calculus . . . . . . . . . . . . . Interval Analysis . . . . . . . . . . . . . . . 1.7.1 Interval arithmetic . . . . . . . . . . 1.7.2 Integral of interval-valued functions Stochastic Processes . . . . . . . . . . . . .

1 . . . . . . . . . .

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2 Integral Inequalities for Strongly Generalized Convex Functions 2.1 2.2 2.3

2.4 2.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . Hermite–Hadamard Type Inequalities for Functions Whose Derivatives are Strongly η-Convex . . . . . . . . . . . . . . . 2.3.1 Application to special means . . . . . . . . . . . . . . Weighted Version of Hermite–Hadamard Type Inequalities for Strongly GA-Convex Functions . . . . . . . . . . . . . . . . . Hermite–Hadamard Type Integral Inequalities for the Class of Strongly Convex Functions on Time Scales . . . . . . . . . .

1 5 9 10 12 14 16 16 17 19 21 21 22 28 40 44 52

v

vi

Contents

3 Integral Inequalities for Strongly Generalized Convex Functions of Higher Order 3.1 3.2 3.3 3.4

67

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . Strongly Generalized Convex Functions of Higher Order . . . Integral Inequalities for Higher Order Strongly Exponentially Convex Functions . . . . . . . . . . . . . . . . . . . . . . . .

4 Integral Inequalities for Generalized Preinvex Functions 4.1 4.2 4.3 4.4 4.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . Hermite–Hadamard Type Inequalities via Preinvex Functions Application to Special Means . . . . . . . . . . . . . . . . . . Generalized (m,h)-Preunivex Mappings via k -Fractional Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Some Majorization Integral Inequalities for Functions Defined on Rectangles via Strong Convexity 5.1 5.2 5.3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . Majorization Integral Inequalities for Strong Convexity . . .

6.4 6.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . Hermite–Hadamard Type Inclusions for Interval-Valued Preinvex Functions . . . . . . . . . . . . . . . . . . . . Hermite–Hadamard Type Inclusions for Interval-Valued Coordinated Preinvex Functions . . . . . . . . . . . . . Hermite–Hadamard Type Fractional Inclusions for Harmonically h-Preinvex Interval-Valued Functions . .

77 93 93 94 96 106 108 133

6 Hermite–Hadamard Type Inclusions for Interval-Valued Generalized Preinvex Functions 6.1 6.2 6.3

67 68 69

133 134 135 149

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149 150

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151

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161

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177

7 Some Inequalities for Multidimensional General h-Harmonic Preinvex and Strongly Generalized Convex Stochastic Processes 191 7.1 7.2 7.3

Introduction . . . . . . Preliminaries . . . . . . General h−Harmonically (Gh − HPηϕ SP ) . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preinvex Stochastic Process . . . . . . . . . . . . . . . . . . . . .

191 192 196

Contents 7.4 7.5

Multidimensional General h−Harmonic Preinvex Stochastic Processes (M Gh − HPη ϕSP ) . . . . . . . . . . . Strongly Generalized Convex Stochastic Processes . . . . . .

8 Applications 8.1

8.2 8.3 8.4

Hermite–Hadamard Inequality . . . . . . . . . . . . . . . . . 8.1.1 Higher dimensional Hermite–Hadamard inequality . . 8.1.2 Mass transportation and higher dimensional Hermite–Hadamard inequality . . . . . . . . . . . . . Jensen’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . Interval-Valued Functions . . . . . . . . . . . . . . . . . . . . 8.4.1 Error estimation to quadrature rules Using Ostrowski type inequality for interval-valued functions . . . . . .

vii 206 216 231 231 232 233 234 236 237 238

Bibliography

241

Index

257

Foreword

Integral inequalities constitute a very important topic in mathematics, in view of their applications in the calculus of variations, differential geometry, mathematical analysis, number theory, differential equations, probability theory and statistics, among other fields. On the other hand, generalized convexity is a set of concepts and techniques that provide useful tools in many branches of applied sciences, including economics, engineering, finance, mechanics, variational analysis and, very specially, optimization theory. Even though some articles have explored connections between these two topics, the book “Integral Inequalities and Generalized Convexity”, by Shashi Kant Mishra, Nidhi Sharma and Jaya Bisht, is the first monograph devoted to a systematic exploration of the relationship existing between them. It shows the usefulness of generalized convexity in obtaining both classical and new integral inequalities. The fundamental generalized convexity notion considered in the book is that of η-invexity, a very natural generalization of convexity that consists in replacing x –y in the classical sub-gradient inequality with a suitable function η(x,y). The relevance of this notion stems from the fact that a differentiable function is η-invex for some η if and only if every stationary point is a global minimum. This book makes an extensive use of this notion as well as of the many generalizations thereof that have been proposed in the literature with the aim of treating non-convex problems with convexity type techniques. One interesting feature of “Integral Inequalities and Generalized Convexity” is its use of fractional calculus, a branch of differential calculus that has been widely used in applied mathematics and has given rise to the field of fractional differential equations, an extension of the theory of classical differential equations. Another distinctive feature of “Integral Inequalities and Generalized Convexity” is the consideration of integral inclusions for interval-valued functions. Interval analysis provides a suitable framework for dealing with rounding errors. Since such errors are inevitably present in practical computations, they necessarily have to be considered in every application of integral inequalities to real-world problems, hence the importance of integral inclusions for interval-valued functions. The last chapter of the book provides an excellent motivation to its contents, as it describes several important applications, including an interesting mass transportation interpretation of the Hermite–Hadamard inequality. It

ix

x

Foreword

contains the most recent results on the applications of generalized convexity to integral inequalities. The book “Integral Inequalities and Generalized Convexity” is a valuable addition to the existing literature on integral inequalities. It will be useful for graduate students in mathematics and its applications will undoubtedly become a helpful reference for researchers working in this field. Juan Enrique Mart´ınez-Legaz Emeritus Professor Universitat Aut`onoma de Barcelona, Spain

Author Biographies

Shashi Kant Mishra PhD, DSc is a Professor at the Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, India, with over 24 years of teaching experience. He has authored eight books, including textbooks and monographs, and has been on the editorial boards of several important international journals. He has guest-edited special issues of the Journal of Global Optimization; Optimization Letters (both Springer Nature) and Optimization (Taylor & Francis). He has received INSA Teacher Award 2020 from Indian National Science Academy, New Delhi and DST Fast Track Fellow 2001 from Ministry of Science and Technology, Government of India. Prof. Mishra has published over 203 research articles in reputed international journals and supervised 21 PhD students. He has visited around 15 institutes/ universities in countries such as France, Canada, Italy, Spain, Japan, Taiwan, China, Singapore, Vietnam and Kuwait. His current research interest includes mathematical programming with equilibrium, vanishing and switching constraints, invexity, multiobjective optimization, non-linear programming, linear programming, variational inequalities, generalized convexity, integral inequalities, global optimization, non-smooth analysis, convex optimization, non-linear optimization and numerical optimization. Nidhi Sharma is a Fellow of the Council of Scientific Industrial Research (CSIR) at the Department of Mathematics, Institute of Science, Varanasi, India. She received MSc degree in Mathematics from Banaras Hindu University, Varanasi, India. She is working on generalized convexity and integral inequalities under the supervision of Prof. S. K. Mishra. Her current research interests include integral inequalities, generalized convexities, set-valued functions and stochastic processes.

xi

xii

Author Biographies

Jaya Bisht is a DST-INSPIRE Fellow (Senior Research Fellow) at the Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, India. She received BSc and MSc degrees in Mathematics from Hemwati Nandan Bahuguna Garhwal University (Central University), Srinagar, India. She is awarded Junior Research Fellowship (JRF) from Human Resource Development Group of Council of Scientific Industrial Research (HRDCSIR), Government of India. Her current research interest includes integral inequalities, generalized convexities, mathematical analysis, interval-valued functions and convex stochastic processes.

Preface

The generalization of convex functions is considered as an original icon in the theoretical study of mathematical inequalities. Integral inequalities involving generalized convexity play an important role in many branches of mathematics and have recently drawn the attention of a large number of researchers interested in both theory and applications. The subject has received tremendous impetus from outside of mathematics from such diverse fields as mathematical economics, game theory, mathematical programming, control theory, variational methods, information theory, probability theory and statistics. It is recognized that some specific inequalities, such as Hermite–Hadamard inequality, Jensen inequality, Ostrowski inequality and Majorization inequality provide a useful and important device in the development of various branches of mathematics. Because of the abundance of applications, the theory of these inequalities is rapidly developing, and it is currently one of the most rapidly developing areas of mathematics. Researchers in various branches of mathematics have discovered generalizations, extensions, refinements, improvements, discretizations, and new applications of these inequalities using generalized convexity over the years. These developments have inspired the authors to write a monograph devoted to the most recent results in mathematics related to these most important integral inequalities. The monograph covers several new research findings in the area of generalized convexity and integral inequalities. The material of the book concentrated on generalization and extension of the classical convexity in different directions and integral inequalities for these generalized and extended convexity with basic knowledge of calculus, measure theory and real analysis. The book will be useful for graduates and researchers who are working in the field of generalized convexity and integral inequalities. Furthermore, the book can serve as a valuable reference text for anyone including experts working in this research field. The book is organized as follows: it has eight chapters; Chapter 1 is introductory and contains basic definitions and concepts needed in the book. Chapter 2 presents integral inequalities for strongly generalized convex functions. We establish some Hermite–Hadamard and Fej´er-type inequalities for strongly η-convex functions. We discuss some applications to special means of real numbers with the help of these results. Further, we establish some new weighted Hermite–Hadamard inequalities for strongly GA-convex functions by using geometric symmetry of a continuous positive mapping and a xiii

xiv

Preface

differentiable mapping whose derivatives in absolute value are strongly GAconvex. Furthermore, we introduce the notion of a strongly convex function on time scales and derive some new dynamic inequalities for these strongly convex functions. Chapter 3 contains integral inequalities for strongly generalized convex functions of higher order. We introduce the concept of strongly η-convex functions of higher order and investigate the Hermite–Hadamard and Hermite– Hadamard-Fej´er type inequalities for these functions. Further, we derive Hermite–Hadamard and related integral inequalities for higher order strongly exponentially convex functions. We also discuss Reimann-Liouville fractional estimates via strongly exponentially convex functions of higher order. Chapter 4 contains integral inequalities for generalized preinvex functions. We prove a new form of Hermite–Hadamard inequality using left and rightsided ψ-Riemann–Liouville fractional integrals for preinvexity. We present two essential results of ψ-Riemann–Liouville fractional integral identities using the first-order derivative of a preinvex function. Further, we propose the concept of generalized (m, h)-preunivex functions and establish some new bounds on Hermite–Hadamard and Simpson’s inequalities for mappings whose absolute values of second derivatives are generalized (m, h)-preunivex. Chapter 5 deals with majorization integral inequalities for strongly convex functions defined on rectangles. We extend several integral majorization type and generalized Favard’s inequalities from functions defined on intervals to functions defined on rectangles via strong convexity and apply the results to establish some new integral inequalities for functions defined on rectangles. Chapter 6 presents Hermite–Hadamard type inclusions for generalized preinvex interval-valued functions. We introduce the concept of (h1 , h2 )preinvex interval-valued functions, coordinated preinvex interval-valued functions, and harmonically h-preinvex interval-valued functions. Further, we establish inclusions of Hermite–Hadamard type for preinvex and coordinated preinvex interval-valued functions. Furthermore, We prove Hermite– Hadamard type inclusions for harmonically h-preinvex interval-valued functions by using interval-valued Riemann–Liouville fractional integrals. Chapter 7 presents integral inequalities for generalized convex stochastic processes. We define general h−harmonic preinvex stochastic processes and the multidimensional general h−harmonic preinvex stochastic processes. We prove the Hermite–Hadamard inequality and obtain some important results for these stochastic processes. Further, we introduce the concept of strongly η-convex stochastic processes and obtain the Hermite–Hadamard inequality, Ostrowski inequality and some other interesting inequalities for strongly ηconvex stochastic processes. Chapter 8 contains several applications of Hermite–Hadamard inequality, Jensen’s inequality, time-scale analysis and Ostrowski integral inequality for interval-valued functions. The list of applications related to integral inequalities and generalized convexity is nearly endless, and we are confident that many new and beautiful

Preface

xv

applications will be developed in the future. A detailed and comprehensive account of typical applications may be found in the various references given at the end. An undertaking of this kind cannot be completed without some mistakes. Although we have made every effort to ensure that there are no inaccuracies, since we provide the exact references, we sincerely hope that the reader will forgive the occasional mistake and feel free to study the original reference. The authors are thankful to Ms Isha Singh from CRC Press for her patience, support and effort in handling the book.

Symbol Description

N N0 R R\{0} R R+ R− Rn (c, d) [c, d] φ ∀ int(K)/K 0 epi(ξ) Lα |.| · ·, · ξ  (x) ξ  (x) β(c, d) Γ(.) L1 [c, d]

set of all natural numbers set of all natural numbers included zero set of all real numbers set of all real numbers excluded zero extended real line set of all nonnegative real numbers set of all nonpositive real numbers Euclidean n-space open line segment joining c and d closed line segment joining c and d empty set for all interior of a set K epigraph of a function ξ lower level set absolute value Euclidean norm Euclidean inner product derivative of ξ at x

T

Time scale

[c, d]T

Time-scaled interval

A(c, d)

Arithmetic mean

G(c, d)

Geometric mean

H(c, d)

Harmonic mean

Ln (c, d)

Generalized logarithmic mean

I(c, d)

Identric mean

Hw,m (c, d) Heronian mean RI

set of all closed intervals of R

R+ I

set of all positive closed intervals of R

R− I

set of all negative closed intervals of R

R− I

set of all negative closed intervals of R

R− I

set of all negative closed intervals of R

R([c,d])

collection of all Riemann integrable functions on [c, d]

IR([c,d])

collection of all intervalRiemann integrable functions on [c, d]

second derivative of ξ at ID(∆) x Beta function Gamma function Kc set of all Lebesgue measurable functions on [c, d]

collection of all interval double integrable functions on ∆ family of all non-empty compact convex subsets of R xvii

Chapter 1 Introduction

1.1

Generalized Convexity

Let Rn be the n-dimensional Euclidean space. We denote the usual inner product by ·, · and for x ∈ Rn , · denote the norm defined by x =



n  i=1

xi

2

1/2

.

It is basic knowledge in mathematical analysis that a nonempty subset K of Rn is said to be convex, if and only if for any x, y ∈ K and δ ∈ [0, 1], one has x + δ(y − x) ∈ K. Definition 1.1.1. Let K ⊆ Rn be a nonempty convex set and let ξ : K → R. (a) The function ξ is said to be convex on K if and only if for any x, y ∈ K and δ ∈ [0, 1], one has ξ(δx + (1 − δ)y) ≤ δξ(x) + (1 − δ)ξ(y). (b) The function ξ is said to be strictly convex on K if and only if for any x, y ∈ K and δ ∈ [0, 1], one has ξ(δx + (1 − δ)y) < δξ(x) + (1 − δ)ξ(y). A real-valued function ξ defined on a convex set K ⊆ Rn is concave if and only if −ξ is convex on K. Definition 1.1.2. Let ξ be defined on a convex set K ⊆ Rn . Then (a) The function ξ is said to be quasiconvex on K if and only if for any x, y ∈ K and δ ∈ [0, 1], one has ξ(δx + (1 − δ)y) ≤ max{ξ(x), ξ(y)}. DOI: 10.1201/9781003408284-1

1

2

Integral Inequalities and Generalized Convexity (b) The function ξ is said to be strictly quasiconvex on K if and only if for any x, y ∈ K and δ ∈ [0, 1], one has ξ(δx + (1 − δ)y) < max{ξ(x), ξ(y)}.

Definition 1.1.3. Let K ⊆ R+ = (0, ∞). The set K is said to be GA-convex set, if xδ y 1−δ ∈ K, ∀x, y ∈ K, δ ∈ [0, 1]. Definition 1.1.4. A function ξ : K ⊆ R+ = (0, ∞) → R is said to be GAconvex on K, if ξ(xδ y 1−δ ) ≤ δξ(x) + (1 − δ)ξ(y),

∀x, y ∈ K, δ ∈ [0, 1],

where xδ y 1−δ and δξ(x) + (1 − δ)ξ(y) are the weighted geometric mean of two positive numbers x and y and the weighted arithmetic mean of ξ(x) and ξ(y), respectively. Definition 1.1.5. A function ξ : K ⊆ R+ = (0, ∞) → R is said to be geometrically quasiconvex on K if ξ(xδ y 1−δ ) ≤ sup{ξ(x), ξ(y)}, ∀x, y ∈ K and δ ∈ [0, 1]. The following concepts of s-convex, tgs-convex and MT-convex are given by Hudzik and Maligranda [62] and Tunc et al. [167, 168], respectively. Definition 1.1.6. A function ξ : [0, ∞) → R is named s-convex in the second sense along with s ∈ (0, 1] if ξ(αx + βy) ≤ αs ξ(x) + β s ξ(y) holds for all x, y ∈ [0, ∞) and α, β ≥ 0 along with α + β = 1. Definition 1.1.7. A function ξ : K ⊆ R → R is named tgs-convex on X if ξ is non negative and ξ(δx + (1 − δ)y) ≤ δ(1 − δ)[ξ(x) + ξ(y)] holds for all x, y ∈ K and δ ∈ (0, 1). Definition 1.1.8. A function ξ : K ⊆ R → R is called M T -convex if ξ is non-negative and √ √ δ 1−δ ξ(x) + √ ξ(y) ξ(δx + (1 − δ)y) ≤ √ 2 1−δ 2 δ holds for all x, y ∈ K and δ ∈ (0, 1).

Introduction

3

Karamardian [72] gave the definition of strongly convex function. Definition 1.1.9. A function ξ : K ⊆ Rn −→ R is said to be strongly convex on a convex set K ⊆ Rn if there exists a constant µ > 0 such that ξ(δx + (1 − δ)y) ≤ δξ(x) + (1 − δ)ξ(y) − µδ(1 − δ)y − x2

(1.1)

for any x, y ∈ K and δ ∈ [0, 1]. The following theorem shows that the definition of the convex functions may be extended to any weighted average of its values at a finite number of points. Theorem 1.1.1. (Jensen Inequality) Let K ⊆ Rn be a nonempty convex set and let ξ : K → R. (a) The function ξ is convex on K if and only if for any x1 , x2 , ..., xn ∈ K and δi ≥ 0, i = 1, 2, ..., n, one has  n  n n    ξ δi xi ≤ δi ξ(xi ), δi = 1. i=1

i=1

i=1

(b) The function ξ is strictly convex on K if and only if for any x1 , x2 , ..., xn ∈ K and δi ≥ 0, i = 1, 2, ..., n, one has  n  n n    ξ δi xi < δi ξ(xi ), δi = 1. i=1

i=1

i=1

Definition 1.1.10. The epigraph of any function ξ is given by epi ξ = {(x, α) : x ∈ K, ξ(x) ≤ α, α ∈ R}. In the following theorem the convex theorem is characterized by the convexity of its epigraph. Theorem 1.1.2. Let K ⊆ Rn be a nonempty convex set and let ξ : K → R. Then ξ is convex on K if and only if epi ξ is convex set. The lower-level set of any function ξ : K → R is given by Lα = {x ∈ K : ξ(x) ≤ α}. The convexity of the lower-level set is a necessary condition for a function to be convex. Theorem 1.1.3. Let K ⊆ Rn be a nonempty convex set and let ξ : K → R. If ξ is convex on K, then Lα is convex for any α ∈ R. The following theorem gives the algebraic structure of the convex functions.

4

Integral Inequalities and Generalized Convexity

Theorem 1.1.4. Let ξ1 , ξ2 , ..., ξm be real-valued functions defined on a m nonempty convex set K ⊆ Rn and let ξ(x) = i=1 αi ξi (x), αi ≥ 0. Then (a) If ξi , i = 1, 2, ..., m are convex on K, then ξ is convex on K.

(b) If ξi , i = 1, 2, ..., m are strictly convex on K, then ξ is strictly convex on K. The following theorem related to the composition of functions is important in the construction of convex functions. Theorem 1.1.5. Let ξ : K → R be a convex function defined on a convex set K ⊆ Rn and let ψ : A → R be a nondecreasing convex function, with ξ(K) ⊆ A. Then, the composite function φ(x) = ψ(ξ(x)) is convex on K. Furthermore, if ξ is strictly convex and ψ is an increasing convex function, then φ is strictly convex. The following theorems characterize differentiable convex and strongly convex functions. Theorem 1.1.6. Let ξ be a differentiable function defined on a nonempty open convex set K ⊆ Rn . Then ξ is convex on K if and only if for every x, y ∈ K, one has ξ(x) − ξ(y) ≥ ∇ξ(y), x − y.

Theorem 1.1.7. Let ξ be a differentiable function defined on a nonempty open convex set K ⊆ Rn . Then ξ is strongly convex on K with modulus µ > 0 if and only if for every x, y ∈ K, one has ξ(x) − ξ(y) ≥ ∇ξ(y), x − y + µx − y2 .

The following concepts of coordinate convex functions are given by Dragomir [45]. Definition 1.1.11. A function ξ : ∆ = [a, b] × [c, d] → R is said to be coordinate convex if the partial mappings ξy : [a, b] → R defined as ξy (u) = ξ(u, y) for all y ∈ [c, d] and ξx : [c, d] → R defined as ξx (v) = ξ(x, v) for all x ∈ [a, b] are convex. Lemma 1.1.1. Every convex function defined on a rectangle is coordinate convex, but the converse is not true, in general.

Example 1.1.1. The mapping ξ : [0, 1]2 → [0, ∞) given by ξ(x, y) = xy. Then ξ is convex on the coordinates but not convex on [0, 1]2 . The following concepts of coordinate strongly convex functions are given by Khan et al. [84]. Definition 1.1.12. A function ξ : ∆ = [a, b] × [c, d] → R is said to be coordinate strongly convex if the partial mappings ξy : [a, b] → R defined as ξy (u) = ξ(u, y) for all y ∈ [c, d] and ξx : [c, d] → R defined as ξx (v) = ξ(x, v) for all x ∈ [a, b] are strongly convex.

Lemma 1.1.2. Every strongly convex function ψ : ∆ = [a, b] × [c, d] → R is coordinate strongly convex, but the converse is not true in general.

Introduction

1.2

5

Invexity

Hanson [57] introduced a new class of functions which was later termed as the class of invex functions by Craven [33]. Definition 1.2.1. Given a nonempty open subset K of Rn , a mapping η : K × K → Rn and a differentiable scalar function ξ : K → R. The function ξ is said to be invex at y ∈ K, if and only if for all x ∈ K, one has ξ(x) − ξ(y) ≥ ∇ξ(y), η(x, y). Example 1.2.1. Every real-valued convex function is invex for η(x, y) = x−y. The following important characterization of invexity was first given by Craven and Glover [34] and later Ben-Israel and Mond [16] provided a simple proof of the result. Theorem 1.2.1. ξ is invex if and only if every stationary point is a global minimum. Remark 1.2.1. If ξ has no stationary points, then ξ is invex. Ben-Israel and Mond [16] gave the concept of invex sets as follows: Definition 1.2.2. The set K ⊆ Rn is said to be invex with respect to vector function η : Rn × Rn → Rn , if y + δη(x, y) ∈ K,

∀x, y ∈ K,

δ ∈ [0, 1].

It is well known that every convex set is invex with respect to η(x, y) = x − y, but not conversely. Remark 1.2.2. (a) The concept of invex set is a generalization of the notion of convex set. (b) Every set in Rn is an invex set with respect to η(x, y) = 0, ∀x, y ∈ Rn . (c) The only function ξ : Rn → R invex with respect to η(x, y) = 0 is the constant function ξ(x) = c, c ∈ R. Ben-Israel and Mond [16] and Weir and Jeyakumar [172] studied the class of preinvex functions to relax the differentiability requirements in invexity given as follows: Definition 1.2.3. Let K ⊆ Rn be invex with respect to η : K × K → Rn . A function ξ : K → R is said to be preinvex if and only if for any x, y ∈ K and δ ∈ [0, 1], one has ξ(y + δη(x, y)) ≤ δξ(x) + (1 − δ)ξ(y).

6

Integral Inequalities and Generalized Convexity

It is well known that every convex function is a preinvex with respect to η(x, y) = x − y, but not conversely. Ben-Israel and Mond [16] obtained that a differentiable preinvex function is also invex as follows: Proposition 1.2.1. Let K ⊆ Rn be an open invex set with respect to η : K × K → Rn and let ξ : K → R be differentiable on K. If ξ is preinvex with respect to η, then ξ is invex with respect to η. The converse of the above result does not hold in general. Example 1.2.2. The function ξ(x) = ex , x ∈ R is invex with respect to η(x, y) = −1, but not preinvex with respect to the same η. Mohan and Neogy [108] gave the following Condition C imposed on η, a differentiable function which is invex on K, with respect to η, is also preinvex. Definition 1.2.4. Let K ⊆ Rn be an open invex subset with respect to η : K × K → R. The function η satisfies the Condition C if for any x, y ∈ K and any δ ∈ [0, 1], η(y, y + δη(x, y)) = −δη(x, y), η(x, y + δη(x, y)) = (1 − δ)η(x, y). Note that ∀x, y ∈ K and δ ∈ [0, 1], then from Condition C, we have η(y + δ2 η(x, y), y + δη(x, y)) = (δ2 − δ)η(x, y). Example of a set which is not convex but invex is given as follows: Example 1.2.3. The set R\{0} is invex with respect to η satisfying Condition C given by   x − y, x ≥ 0, y ≥ 0 η(x, y) = y − x, x ≤ 0, y ≤ 0   −y, otherwise.

Definition 1.2.5. At y ∈ K, where K is a convex set, a function ξ is said to be B-vex with respect to b, if for every x ∈ K and 0 ≤ δ ≤ 1, ξ(δx + (1 − δ)y) ≤ δb(x, y, δ)ξ(x) + (1 − δb(x, y, δ))ξ(y) = ξ(y) + δb(x, y, δ)(ξ(x) − ξ(y)). Definition 1.2.6. At y ∈ K, where K is a invex set, the function ξ is said to be pre-quasiunivex with respect to η, φ and b, if for every x ∈ K and 0 ≤ δ ≤ 1, φ(ξ(x) − ξ(y)) ≤ 0 =⇒ bξ(y + δη(x, y)) ≤ bξ(y). Bector et al. [15] introduced preunivex and univex functions with respect to a vector function η, scalar function φ and b as generalizations of preinvex, invex, b-preinvex, b-invex and b-vex functions.

Introduction

7

Definition 1.2.7. Given K ⊆ Rn × R. K is said to be univex set with respect to η, φ and b, if every (x, α), (y, β) ∈ K and 0 ≤ δ ≤ 1 =⇒ (y + δη(x, y), β + δbφ(α − β)) ∈ K. Definition 1.2.8. At y ∈ K, the function ξ is said to be univex with respect to η, φ and b, if for every x ∈ K there exists a function b(x, y) such that b(x, y)φ[ξ(x) − ξ(y)] ≥ η(x, y)T ∇ξ(y). Example 1.2.4. If ξ : R → R be defined by ξ(x) = x3 , where  2 y , x > y, b(x, y) = x−y 0, x ≤ y, and η(x, y) =



x2 + y 2 + xy, x − y,

x > y, x ≤ y.

Let φ : R → R be defined by φ(c) = 3c. Then function ξ is univex. Remark 1.2.3. Every invex function is univex, where φ : R → R can be defined by φ(c) = c, b(x, y) ≡ 1, but not conversely. Example 1.2.5. The function considered in Example 1.2.4 is univex but not invex because for x = −3, y = 1, (ξ(x) − ξ(y)) < η(x, y)T ∇ξ(y). Remark 1.2.4. Every convex function is univex, where φ : R → R can be defined by φ(c) = c, b(x, y) ≡ 1 and η(x, y) = x − y, but not conversely. Example 1.2.6. The function considered in Example 1.2.4 is univex but not convex because for x = −2, y = 1, (ξ(x) − ξ(y)) < η(x, y)T ∇ξ(y). Remark 1.2.5. Every B-vex function is univex, where φ : R → R can be defined by φ(c) = c, and η(x, y) = x − y, but not conversely. Example 1.2.7. The function considered in Example 1.2.4 is univex but not 1 1 B-vex because for x = 10 , y = 100 , b(x, y)(ξ(x) − ξ(y)) < η(x, y)T ∇ξ(y). Definition 1.2.9. if ξ : K → R is differentiable and prequasiunivex with respect to η, φ and b then ξ is quasiunivex with respect to η, φ and b, where b(x, u) = limδ→0+ b(x, u, δ). Remark 1.2.6. Every univex function with respect to η, φ and b is quasiunivex with respect to same η, φ and b. However, the converse does not hold, as it shown by the following example.

8

Integral Inequalities and Generalized Convexity

Example 1.2.8. ξ : R → R be defined by ξ(x) = −x2 ,  1, x = −y, b(x, y) = 0, otherwise, and η(x, y) =



y − x, x − y,

x = −y, otherwise,

and φ : R → R is defined by φ(c) = 2c. The function ξ is quasiunivex but not univex, because for x = 1, y = 2, b(x, y)φ(ξ(x) − ξ(y)) < η(x, y)T ∇ξ(y). Matloka [104] gave the following concept of invex set and preinvex functions on the coordinates. Let K1 and K2 be two nonempty subsets of Rn , let η1 : K1 × K1 → Rn and η2 : K2 × K2 → Rn . Definition 1.2.10. Let (u, v) ∈ K1 × K2 . The set K1 × K2 is said to be invex at (u, v) with respect to η1 and η2 , if for each (x, y) ∈ K1 × K2 and λ1 ,λ2 ∈ [0, 1], (u + λ1 η1 (x, u), v + λ2 η2 (y, v)) ∈ K1 × K2 . K1 × K2 is said to be invex set with respect to η1 and η2 if K1 × K2 is invex at each (u, v) ∈ K1 × K2 . Definition 1.2.11. A non-negative function ξ on the invex set K1 ×K2 is said to be coordinated preinvex with respect to η1 and η2 if the partial mappings ξy : K1 → R defined as ξy (u) = ξ(u, y) for all y ∈ K2 and ξx : K2 → R defined as ξx (v) = ξ(x, v) for all x ∈ K1 are preinvex with respect to η1 and η2 , respectively. Remark 1.2.7. If η1 (x, u) = x − u and η2 (y, v) = y − v, then the function ξ is called convex on the coordinates. The following concept of harmonic convex functions are given by Anderson et al. [5] and I¸scan [66]. Definition 1.2.12. A set K = [c, d] ⊆ R\{0} is called a harmonic convex set, if xy ∈ K, ∀x, y ∈ K, δ ∈ [0, 1]. δx + (1 − δ)y

Definition 1.2.13. A function ξ : K = [c, d] ⊆ R\{0} → R is called harmonic convex, if   xy ≤ (1 − δ)ξ(x) + δξ(y), ∀x, y ∈ K, δ ∈ [0, 1]. ξ δx + (1 − δ)y

Introduction

9

The concept of harmonic invex set and harmonic preinvex function was first given by Noor et al. [126]. Definition 1.2.14. A set K = [c, c + η(d, c)] ⊆ R\{0} is said to be harmonic invex set with respect to the bifunction η(., .), if x(x + η(y, x)) ∈ K, ∀ x, y ∈ K, δ ∈ [0, 1]. x + (1 − δ)η(y, x)

(1.2)

It is well known that every harmonic convex set is harmonic invex with respect to η(y, x) = y − x but not conversely.

Definition 1.2.15. A function ξ : K = [c, c + η(d, c))] ⊆ R\{0} → R is said to be harmonic preinvex with respect to the bifunction η(., .), if   x(x + η(y, x)) ≤ (1 − δ)ξ(x) + δξ(y), ∀x, y ∈ K, δ ∈ [0, 1]. ξ x + (1 − δ)η(y, x) Noor et al. [121] defined the relative harmonic preinvex functions. Definition 1.2.16. Let h : [0, 1] ⊆ J → R be a nonnegative function. A function ξ : K = [c, c + η(d, c)] ⊆ R\{0} → R is a relative harmonic preinvex function with respect to an arbitrary nonnegative function h and an arbitrary bifunction η(., .), if   x(x + η(y, x)) ≤ h(1 − δ)ξ(x) + h(δ)ξ(y), ∀x, y ∈ K, δ ∈ [0, 1]. ξ x + (1 − δ)η(y, x) (1.3)

1.3

Integral Inequalities

The classical Hermite–Hadamard type inequality provides lower and upper estimates for the integral average of any convex function defined on a compact interval, involving the midpoint and the endpoints of the domain. This interesting inequality was first discovered by Hermite in 1883 in the journal Mathesis (see [56, 107]). However, this beautiful result was nowhere mentioned in the mathematical literature and was not widely known as Hermite’s result (see [133]). Theorem 1.3.1. (Hermite–Hadamard inequality) Let ξ : [c, d] → R be a convex function with c < d. Then,    d 1 ξ(c) + ξ(d) c+d ≤ . ξ ξ(x)dx ≤ 2 d−c c 2 The following double inequality is known as Hermite–Hadamard Fej´er inequality [50] in the literature:

10

Integral Inequalities and Generalized Convexity

Theorem 1.3.2. (Hermite–Hadamard Fej´er inequality) Let ξ : [c, d] → R be a convex function with c < d. Then  d   d  1 ξ(c) + ξ(d) d c+d ψ(x)dx ≤ ξ(x)ψ(x)dx ≤ ξ(x)dx, ξ 2 d−c c 2 c c where ψ : [c, d] → R is non-negative, integrable, and symmetric about x =

c+d 2 .

Theorem 1.3.3. (Simpson’s inequality) Let ξ : [c, d] → R be a four times continuously differentiable mapping on (c, d) and ξ (4) ∞ = supx∈(c,d) |ξ (4) (x)| < ∞. Then the following inequality holds:       d   1 ξ(c) + ξ(d) c+d 1 1   + 2ξ − ξ (4) ∞ (d − c)4 . f (x)dx ≤   2880 3 2 2 d−c c

Theorem 1.3.4. (Ostrowski inequality) Let ξ : I ⊂ [0, ∞) → R be a differentiable mapping on the interior of the interval I, such that ξ  ∈ L[c, d], where c, d ∈ I with c < d. If |ξ  (x)| ≤ M, then the following inequality holds:      d   1 M (x − c)2 + (d − x)2   . ξ(x)dx ≤ ξ(x) −   d−c d−c c 2

Theorem 1.3.5. (Trapezoid type inequality) Let ξ : int(I) ⊆ R → R be a differentiable mapping on int(I), c, d ∈ int(I) with c < d. If |ξ  | is convex on [c, d], then the following inequality holds:    d ξ(c) + ξ(d)  1  ξ(x)dx − (d − c)   ≤ (d − c)2 (|ξ  (c)| + |ξ  (d)|).  c  8 2

(1.4)

Theorem 1.3.6. (Mid-point type inequality) Let ξ : int(I) ⊆ R → R be a differentiable mapping on int(I), c, d ∈ int(I) with c < d. If |ξ  | is convex on [c, d], then the following inequality holds:     d c + d  1  ξ(x)dx − (d − c)ξ (1.5)   ≤ (d − c)2 (|ξ  (c)| + |ξ  (d)|).  c  8 2

1.4

Fractional Calculus

In recent years, fractional calculus has been proven a powerful tool for the study of dynamical properties of many interesting systems in physics, chemistry and engineering. It serves as a powerful application in nonlinear oscillations of earthquakes, many physical phenomena such as seepage flow

Introduction

11

in porous media and in fluid dynamical traffic modelling. For more recent developments on fractional calculus and application to fractional differential equations, refer to [14, 171] and references therein. Now, we shall discuss the following special functions: The Gamma function  ∞ Γ(α) = e−δ δ (α−1) dδ. 0

The Beta function: Γ(x)Γ(y) = β(x, y) = Γ(x + y)



1

δ (x−1) (1 − δ)(y−1) dδ, x, y > 0.

0

The incomplete Beta function:  a β(a; x, y) = δ (x−1) (1 − δ)(y−1) dδ, 0 < a < 1, x, y > 0. 0

The Hypergeometric function: 2 F1 (a, c; d; z)

=

1 β(c, d − c)



0

1

δ (c−1) (1 − δ)(d−c−1) (1 − zδ)−a dδ,

d > c > 0, |z| < 1. Definition 1.4.1. Let ξ ∈ L1 [c, d]. The symbol Jcα+ ξ and Jdα− ξ denote the left-sided and right-sided Riemann–Liouville fractional integrals of the order α ∈ R+ are defined by  x 1 α (x − δ)α−1 ξ(δ)dδ, (0 ≤ c < x < d) Jc+ ξ(x) = Γ(α) c and Jdα− ξ(x) =

1 Γ(α)



d x

(δ − x)α−1 ξ(δ)dδ,

(0 ≤ c < x < d)

respectively, where Γ(.) is the Gamma function. Mubeen and Habibullah [112] introduced the following Riemann–Liouville k−fractional integrals. Definition 1.4.2. Let ξ ∈ L1 [c, d] then the Riemann–Liouville k-fractional integrals k Jcα+ ξ(x) and k Jdα− ξ(x) of order α > 0 are given as  x α 1 α (x − δ) k −1 ξ(δ)dδ, (0 ≤ c < x < d) J ξ(x) = k c+ kΓk (α) c

12

Integral Inequalities and Generalized Convexity

and α k Jd− ξ(x)

1 = kΓk (α)



d x

α

(δ − x) k −1 ξ(δ)dδ,

(0 ≤ c < x < d)

respectively, where k > 0 and Γk (α) is the k-Gamma function defined by  ∞ δk Γk (α) = 0 e− k δ (α−1) dδ. Furthermore, Γk (α + k) = αΓk (α) and k Jc0+ ξ(x) =k Jd0− ξ(x) = ξ(x). Sarikaya et al. [149] studied inequalities of Hermite–Hadamard type involving Riemann–Liouville fractional integrals and produced a new fractional Hermite–Hadamard type inequality as follows: Theorem 1.4.1. Let ξ : [c, d] → R be a positive function along with 0 ≤ c < d, and let ξ ∈ L1 [c, d]. Suppose that ξ is a convex function on [c, d], then the following inequalities for fractional integrals hold:   c+d Γ(α + 1) α ξ(c) + ξ(d) ξ ≤ , [J + ξ(d) + Jdα− ξ(c)] ≤ 2 2(d − c)α c 2 where the symbols Jcα+ ξ and Jdα− ξ denote, respectively, the left-sided and rightsided Riemann–Liouville fractional integrals of order α > 0 defined by  x 1 (x − δ)(α−1) ξ(δ)dδ, c < x, Jcα+ ξ(x) = Γ(α) c and Jdα− ξ(x) =

1 Γ(α)



d x

(δ − x)(α−1) ξ(δ)dδ, x < d,

where Γ(α) is the gamma function and defined as Γ(α) =

1.5

∞ 0

e−δ δ (α−1) dδ.

Majorization Inequalities

The theory of majorization is a very significant topic in mathematics; a remarkable and complete reference on the majorization subject is the book by Marshall et al. [103]. For example, the theory of majorization is an essential tool that permits us to transform nonconvex complicated constrained optimization problems that involve matrix-valued variables into simple problems with scalar variables that can be easily solved [32, 61]. A vector x = (x1 , x2 , ..., xn ) is said to be majorized by a vector m y = , y , ..., y ), in symbol y  x, if x ≥ ... ≥ x , y ≥ ... ≥ y , (y n n i=1 yi ≥ 1m 2 n1 nn 1 x , m = 1, 2, ..., n − 1 and y = x . It means that the sum i=1 i i=1 i i=1 i

Introduction

13

of m largest entries of x does not exceed the sum of m largest entries of y; for all m = 1, 2, ..., n. The following theorem is well-known in the literature as the majorization theorem, and for its proof, we refer to [103]. This result is due to Hardy et al. [58] and it can also be found in [73]. Theorem 1.5.1. Let K be an interval in R, and x = (x1 , x2 , ..., xn ) and y = (y1 , y2 , ..., yn ) be two n-tuples such that xi , yi ∈ K(i = 1, 2, ..., n). Then the inequality n n   ξ(ci ) ≥ ξ(di ) i=1

i=1

holds for every continuous convex function ξ : K → R if and only if c  d.

The following theorem is a weighted version of above theorem and is given by Fuchs [52]. Theorem 1.5.2. Let c = (c1 , c2 , ..., cn ) and d = (d1 , d2 , ..., dn ) be two decreasing n-tuples such that ci , di ∈ K(i = 1, 2, ..., n) and p = (p1 , p2 , ..., pn ) be a real n-tuple with k  i=1

k 

pi d i ,

n 

pi c i ≥

n 

pi d i .

pi ξ(ci ) ≥

n 

pi ξ(di )

pi c i ≥

i=1

i=1

Then the inequality

n  i=1

k = 1, 2, ..., n − 1,

i=1

i=1

holds for every continuous convex function ξ : K → R. The definition of majorization for integrable functions can be stated as follows (see, [133]). Definition 1.5.1. Let f and g be two decreasing real-valued integrable functions on the interval [a, b]. Then f is said to majorize g, in symbol, f  g, if the inequality  x  x g(u)du ≤ f (u)du holds for all x ∈ [a, b) a

and

a



b

g(u)du = a



a

b

f (u)du.

14

1.6

Integral Inequalities and Generalized Convexity

Time Scale Calculus

In 1988, Hilger [60] introduced the theory of time scale which is a unification of the discrete theory with the continuous theory. Recently, much attention has been given to the time scales calculus by many researchers, see for instance [22, 39, 161, 165]. Consequently, the concept of time scale theory has been extended and generalized. Time scales calculus has applications in various fields such as Economics, Engineering, Physics, Signal processing, Aerospace, Dynamic programming, Recurrent neural networks and Control theory, see references [9, 23, 151, 152, 176, 178]. Time scale is a nonempty closed subset of the set of real numbers R. The set of integers Z, the set of real numbers R, finite unions of disjoint intervals, limit sets such as {0} ∪ { n1 } : n = 1, 2, ..., Cantor sets etc are the examples of time scales. We denote time scale by T, time-scaled interval by [c, d]T , and the interior of K by K 0 . There are two types of the operator: The forward jump operator (δ) = inf {x ∈ T : x > δ} and the backward jump operator ς(δ) = sup{x ∈ T : x < δ} for all δ ∈ T. The forward jump operator represents the next element and the backward jump operator represents the previous element in the domain. If T has a maximum δ, then (δ) = δ, and if T has a minimum δ, then ς(δ) = δ. If (δ) > δ, then δ is called right-scattered and if ς(δ) < δ, then δ is called left-scattered. The point δ is said to be isolated if it is both right-scattered and left-scattered: (δ) > δ > ς(δ) for δ ∈ T. It is a characteristic of discrete domains that all points within them are isolated. δ is said to be right-dense if (δ) = δ and δ is said to be left-dense if ς(δ) = δ. The point δ is said to be dense if it is both left-dense and right-dense: (δ) = δ = ς(δ) for δ ∈ T. The mappings η, ζ : T → [0, ∞) defined by η(δ) := (δ) − δ,

ζ(δ) := δ − ς(δ)

are said to be forward and backward graininess functions, respectively. The graininess function measures the step size between two consecutive points in T. The set Tk which is derived from time scale T is defined as follows: If T has a left-scattered maximum m, then Tk = T − m; otherwise Tk = T. The delta derivative is a basic time scale derivative and is denoted by ξ ∆ (δ). Let ξ : T → R be a function. Then the delta derivative ξ ∆ (δ) of ξ at a point δ ∈ Tk is defined to be the number such that given  > 0, there exists a neighbourhood N of δ, such that |ξ((δ)) − ξ(s) − ξ ∆ (δ)((δ) − s)| ≤ |(δ) − s|, for all s ∈ N.

Introduction

15

If T = R, then the delta derivative ξ ∆ = ξ  where ξ  is the derivative from continuous calculus. If T = Z, then the delta derivative ξ ∆ = ∆ξ where ∆ξ is the forward difference operator from discrete calculus. The following definitions and results are given by Bohner and Peterson [22]. Definition 1.6.1. A function ξ : T → R is called rd-continuous if it is continuous at every right-dense point of T and if its left-sided limit is finite at any left-dense point of T. All rd-continuous functions are denoted by Crd . Definition 1.6.2. A function F : T → R is called an antiderivative of ξ : T → R if F ∆ (δ) = ξ(δ), for all δ ∈ Tk . Then, delta integral is defined by  s ξ(δ)∆δ = F (s) − F (c), c

where s, c ∈ T. Theorem 1.6.1. If ξ ∈ Crd and δ ∈ Tk then 

(δ)

ξ(x)∆(x) = η(δ)ξ(δ).

δ

Theorem 1.6.2. Let ξ1 , ξ2 ∈ Crd , λ ∈ R and c, d, α ∈ T then d d d (i) c (ξ1 (x) + ξ2 (x))∆x= c ξ1 (x)∆x + c ξ2 (x)∆x; d d (ii) c λξ(x)∆x = λ c ξ(x)∆x; d c (iii) c ξ(x)∆x = − d ξ(x)∆x; d α d (iv) c ξ(x)∆x = c ξ(x)∆x + α ξ(x)∆x; d d (v) c ξ1 (x)ξ2θ (x)∆x = (ξ1 ξ2 )(d) − (ξ1 ξ2 )(c) − c ξ1θ (x)ξ2 (x)∆x; d d (vi) c ξ1 (x)ξ2θ (x)∆x = (ξ1 ξ2 )(d) − (ξ1 ξ2 )(c) − c ξ1θ (x)ξ2 (x)∆x; c (vii) c ξ(x)∆x = 0; d (viii) If ξ(x) ≥ 0 for all x, then c ξ(x)∆x ≥ 0; d d (ix) If |ξ1 (x)| ≤ ξ2 (x) on [c, d], then | c ξ1 (x)∆x| ≤ c ξ2 (x)∆x.

From assertion (ix) of Theorem 1.6.2 for ξ2 (x) = |ξ1 (x)| on [c, d], we have     d  d   ξ1 (x)∆x ≤ |ξ1 (x)|∆x.   c  c

16

Integral Inequalities and Generalized Convexity

1.7

Interval Analysis

The theory of interval analysis was initiated by Moore [109]. We can compute arbitrarily sharp upper and lower bounds on exact solutions of many problems in applied mathematics by using interval arithmetic, interval-valued functions, integrals of interval-valued functions, etc. Moore [109] supplied techniques for keeping track of errors, developed and applied, for the machine computation of rigorous error bounds on approximate solutions. Computational tests for machine convergence of iterative methods existence and nonexistence of solutions for a variety of equations are obtained via interval analysis. Several researchers focused on studying the literature and applications of interval analysis in automatic error analysis, computer graphics, neural network output optimization, robotics, computational physics, etc.

1.7.1

Interval arithmetic

The rules for interval addition, subtraction, product and quotient [109] are [X, X] + [Y , Y ] = [X + Y , X + Y ]. [X, X] − [Y , Y ] = [X − Y , X − Y ]. X.Y = {xy : x ∈ X, y ∈ Y }.

It is easy to see that X.Y is again an interval, whose ends points can be computed from X.Y = min{X Y , X Y , X Y , X Y } and X.Y = max{X Y , X Y , X Y , X Y }. The reciprocal of an interval as follows 1/X = {1/x : x ∈ X}. If X is an interval not containing the number 0, then 1/X = [1/X, 1/X]. X/Y = X.(1/Y ) = {x/y : x ∈ X, y ∈ Y }, where 1/y is defined by (1.6). Scalar multiplication of the interval X is defined by   [λX, λX], if λ > 0, λX = λ[X, X] = {0}, if λ = 0,   [λX, λX], if λ < 0, where λ ∈ R.

(1.6)

Introduction

17

The Hausdorff distance between X = [X, X] and Y = [Y , Y ] is defined as d(X, Y ) = d([X, X], [Y , Y ]) = max{| X − Y |, | X − Y |}. − Let RI , R+ I and RI be the sets of all closed intervals of R, sets of all positive closed intervals of R and sets of all negative closed intervals of R, respectively. Now, we discuss some algebraic properties of interval arithmetic [109].

(1) (Associativity of addition) (X + Y ) + Z = X + (Y + Z), ∀ X, Y, Z ∈ RI . (2) (Additive element) X + 0 = 0 + X = X, ∀ X ∈ RI . (3) (Commutativity of addition) X + Y = Y + X, ∀ X, Y ∈ RI . (4) (Cancellation law) X + Z = Y + Z =⇒ X = Y, ∀ X, Y, Z ∈ RI . (5) (Associativity of multiplication) (X.Y ).Z = X.(Y.Z), ∀ X, Y, Z ∈ RI . (6) (Commutativity of multiplication) X.Y = Y.X, ∀ X, Y ∈ RI . (7) (Unit element) X.1 = 1.X = X, ∀ X ∈ RI . (8) (Associate law) λ(µX) = (λµ)X, ∀ X ∈ RI and ∀ λ, µ ∈ R. (9) (First distributive law) λ(X +Y ) = λX +λY, ∀ X, Y ∈ RI and ∀ λ ∈ R. (10) (Second distributive law) (λ + µ)X = λX + µX, ∀ X ∈ RI and ∀ λ, µ ∈ R. However, the distributive law does not always hold. Example 1.7.1. X = [−2, −1], Y = [−1, 0] and Z = [1, 3]. X.(Y + Z) = [−2, −1].([−1, 0] + [1, 3]) = [−6, 0] whereas X.Y + X.Z = [−2, −1].[−1, 0] + [−2, −1].[1, 3] = [−6, 1].

1.7.2

Integral of interval-valued functions

A function ξ is said to be an interval-valued function of δ on [c, d] if it assigns a nonempty interval to each δ ∈ [c, d] ξ(δ) = [ξ(δ), ξ(δ)], where ξ and ξ are real-valued functions. A tagged partition P of [a, b] is a set of numbers {ti−1 , ui , ti }m i=1 such that P : a = t0 < t1 < ... < tm = b. with ti−1 ≤ ui ≤ ti for all i = 1, 2, 3 . . . m. Partition P is said to be ρ-fine if ∆ti < ρ for all i, where ∆ti = ti − ti−1 . Let P(ρ, [a, b]) be the family of all

18

Integral Inequalities and Generalized Convexity

ρ-fine partitions of [a, b] . Then, we define the sum S(ξ, P, ρ) =

m  i=1

ξ(ui )[ti − ti−1 ],

where ξ : [c, d] → RI . S(ξ, P, ρ) denotes the Riemann sum of ξ corresponding to the P ∈ P(ρ, [a, b]). The following definition is given by Piatek [134]. Definition 1.7.1. A function ξ : [a, b] → RI is called interval Riemann integrable (IR-integrable) on [a, b] if there exist K ∈ RI such that, for each  > 0, there exist ρ > 0 such that d(S(ξ, P, ρ), K) <  for every Riemann sum S of ξ corresponding to each P ∈ P(ρ, [a, b]) and independent of choice of ui ∈ [ti−1 , ti ] for 1 ≤ i ≤ m.  is called the IR−integral of ξ on [a, b] and is denoted by  b ξ(δ)dδ.  = (IR) a

The collection of all (IR)–integrable functions on [a, b] denoted by IR([a,b]) . Zhao et al. [184] gave the concept of interval double integral for intervalvalued functions as follows: n If P1 = {ti−1 , ui , ti }m i=1 such that P1 ∈ P(ρ, [a, b]) and P2 = {sj−1 , vj , sj }j=1 such that P2 ∈ P(ρ, [c, d]), then the rectangles

∆i,j = [ti−1 , ti ] × [sj−1 , sj ] partition rectangle ∆ = [a, b] × [c, d] with the points (ui , vj ) are inside the rectangles [ti−1 , ti ]×[sj−1 , sj ]. Let P(ρ, ∆) be the family of all ρ-fine partitions P of ∆ such that P = P1 × P2 . Let ∆Ai,j be the area of the rectangle ∆i,j . Choose an arbitrary (ui , vj ) from each rectangle ∆i,j , where 1 ≤ i ≤ m, 1 ≤ j ≤ n, and we get S(ξ, P, ρ, ∆) =

m  n 

ξ(ui , vj )∆Ai,j ,

i=1 j=1

where ξ : ∆ → RI . S(ξ, P, ρ, ∆) denotes integral sum of ξ corresponding to the P ∈ P(ρ, ∆).

Definition 1.7.2. A function ξ : ∆ = [a, b] × [c, d] → RI is called interval   double integrable (ID-integrable) on ∆ with (ID)-integral I = (ID) ∆ ξ(δ1 , δ2 )dA if for each  > 0, there exists ρ > 0 such that d(S(ξ, P, ρ, ∆), I) < 

for each P ∈ P(ρ, ∆).

Introduction

19

The collection of all (ID)-integrable functions on ∆ denoted by ID(∆) . Theorem 1.7.1. If ξ : ∆ = [a, b] × [c, d] → RI be an interval-valued function such that ξ = [ξ, ξ] and ξ ∈ ID(∆) , then we have (ID)

1.8

 

ξ(δ1 , δ2 )dA = (IR) ∆



b

(IR) a



d

ξ(δ1 , δ2 )dδ1 dδ2 .

c

Stochastic Processes

The notion of stochastic processes for convexity is of great importance in optimization and also useful for numerical approximations when there exist probabilistic quantities in the literature [28]. A stochastic process ξ(t) is a function which maps the index set T into the space S of random variable defined on (Ω, A, P ). Sobczyk [164] introduced the concepts of continuity, mean square continuity, monotonicity and differentiability for stochastic processes. Definition 1.8.1. The stochastic process ξ : K × Ω → R is called • continuous in probability in interval K, if for all u0 ∈ K P − lim ξ(u, .) = ξ(u0 , .), u→u0

where P − lim denotes the limit in probability; • mean square continuous in the interval K, if for all u0 ∈ K lim E[(ξ(u, .) − ξ(u0 , .))2 ] = 0,

u→u0

where E[ξ(u, .)] denotes the expectation value of the random variable ξ(u, .); • increasing(decreasing), if for all u, v ∈ K such that u < v, ξ(u, .) ≤ ξ(v, .), (ξ(u, .) ≥ ξ(v, .)); • monotonic if it is increasing or decreasing; • differentiable at a point u0 ∈ K if there is a random variable ξ  (u0 , .) : Ω→R ξ  (u0 , .) = P − lim

u→u0

ξ(u, .) − ξ(u0 , .) . u − u0

20

Integral Inequalities and Generalized Convexity

The definition of mean square integrability for stochasic processes is given by Sobczyk [164]. Definition 1.8.2. Suppose that ξ : K × Ω → R is a stochastic process with E[ξ(u)2 ] < ∞ for all u ∈ K and [c, d] ∈ K, c = u0 < u1 < u2 < ... < un = d is a partition of [c, d] and Θk ∈ [uk−1 , uk ] for all k = 1, 2, ...n. Further, suppose that φ : K × Ω → R be a random variable. Then, it is said to be mean square integrable of the process ξ on [c, d], if for each normal sequence of partitions of the interval [c, d] and for each Θk ∈ [uk−1 , uk ], k = 1, 2, ...n, we have  2  n  lim E  ξ(Θk )∆(uk − uk−1 ) − φ  = 0. n→∞

k=1

Then, we can write

φ(.) =



d

ξ(v, .)dv almost everywhere. c

Also, mean square integral operator is increasing, that is, 

c

d

ξ(u, .)du ≤



where ξ(u, .) ≤ φ(u, .)in [c, d].

c

d

φ(u, .)du almost everywhere,

Chapter 2 Integral Inequalities for Strongly Generalized Convex Functions

2.1

Introduction

Karamardian [72] introduced strongly convex function. However, there are references citing Polyak [136] has introduced strongly convex functions as a generalization of convex functions, see [105, 117]. It is well known that every differentiable function is strongly convex if and only if its gradient map is strongly monotone [72]. Karamardian [72] showed that every bidifferentiable function is strongly convex if and only if its Hessian matrix is strongly positive definite. Az´ ocar et al. [13] derived an appropriate counterpart of the Fej´er inequalities and presented refinement of Hermite–Hadamard inequalities for strongly convex functions. I¸scan [65] established Hermite–Hadamard-Fej´er inequality for fractional integrals. Further, Park [132] obtained new estimates on the generalization of Hermite–Hadamard-Fej´er type inequalities for differentiable functions whose derivatives in absolute value at certain powers are convex. Gordji et al. [54] introduced the concept of η-convex/(ϕ-convex [55]) functions as a generalization of convex functions and obtained the Hermite–Hadamard, Fej´er, Jensen and Slater type inequalities for η-convex functions. Further, Rostamian Delavar and De La Sen [143] gave some applications for Hermite– Hadamard-Fej´er type integral inequalities for differentiable η-convex functions. Further, Awan et al. [12] introduced the notion of strongly η-convex functions and formulated some new integral inequalities of Hermite–Hadamard type for strongly η-convex functions. For more details, one can refer to [35, 43]. The fractional inequalities play an important role in calculating different means for generalized convexity, so researchers are attracting to develop fractional integral inequalities for generalized convexity. Kwun et al. [90] established Hermite–Hadamard and Fej´er-type inequalities and derived fractional integral inequalities for η-convex functions. Further, Yang et al. [177] investigated some Hermite–Hadamard-type fractional integral inequalities for generalized h-convex functions. Niculescu [113] investigated the class of multiplicatively convex functions by replacing the arithmetic mean to the geometric mean. It is well known that every polynomial p(x) with non-negative coefficients is a multiplicatively DOI: 10.1201/9781003408284-2

21

22

Integral Inequalities and Generalized Convexity

convex on [0, ∞). More generally, every real analytic function ξ(x) = ∞ function n c x with non-negative coefficients is a multiplicatively convex function n n=0 on (0, R), where R denotes the radius of convergence [113]. Niculescu [113] showed that a continuous function  ξ : K ⊂ (0, ∞) → [0, ∞) is multiplicatively √ convex if and only if ξ( xy) ≤ ξ(x)ξ(y), ∀x, y ∈ K. Qi and Xi [139] introduced a new concept of geometrically quasi-convex functions and established some integral inequalities of Hermite–Hadamard type for the function whose derivatives are of geometric quasi-convexity [139]. Noor et al. [127] introduced generalized geometrically convex functions and derived some basic inequalities related to generalized geometrically convex functions. Noor et al. [127] also established new Hermite–Hadamard type inequalities for generalized geometrically convex functions. For more details, one can refer to [94, 114, 124, 125, 163, 181]. Recently, Obeidat and Latif [128] established some new weighted Hermite– Hadamard type inequalities for geometrically quasi-convex functions and showed how we can use inequalities of Hermite–Hadamard type to obtain the inequalities for special means. For more details on Hermite–Hadamard inequalities, we refer to [43, 94, 138, 163, 181] and references therein. Dinu [39] obtained Hermite–Hadamard inequality for convex functions on time scales. In 2019, Tahir et al. [165] established some new Hermite– Hadamard type integral inequalities using the concept of time scales. Recently, Rashid et al. [140] investigated the time scales version of two non-negative auxiliary functions for the class of convex functions and obtained several dynamical variants that are essentially based on Hermite–Hadamard inequality. The organization of this chapter is as follows: In Section 2.2, we recall some basic results that are necessary for our main results. In Section 2.3, we establish some Hermite–Hadamard-and Fej´er-type inequalities for strongly ηconvex functions. We derive some integral inequalities for strongly η-convex functions. Further, in Section 2.3.1 we discuss some applications to special means of real numbers with the help of these results. In Section 2.4, we establish some new weighted Hermite–Hadamard inequalities for strongly GAconvex functions by using geometric symmetry of a continuous positive mapping and a differentiable mapping whose derivatives in absolute value are strongly GA-convex. In Section 2.5, we introduce the notion of a strongly convex function with respect to two auxiliary functions ψ1 and ψ2 on Time scales T. Also, we derive some new dynamic inequalities for these strongly convex functions. Some examples are also mentioned in the support of our theory.

2.2

Preliminaries

In this section, we mention some definitions and related results required for this chapter. Gordji et al. [55] introduced the concept of η-convex functions.

Integral Inequalities for Strongly Generalized Convex Functions

23

Definition 2.2.1. A function ξ : K ⊆ R −→ R is said to be η-convex function with respect to η : R × R −→ R, if ξ(δx + (1 − δ)y) ≤ ξ(y) + δη(ξ(x), ξ(y)), ∀x, y ∈ K, δ ∈ [0, 1]. Awan et al. [12] gave the following concept of strongly η-convex functions. Definition 2.2.2. A function ξ : K ⊆ R → R is said to be strongly η-convex function with respect to η : R × R → R and modulus µ > 0 if ξ(δx+(1−δ)y) ≤ ξ(y)+δη(ξ(x), ξ(y))−µδ(1−δ)(x−y)2 , ∀ x, y ∈ K, δ ∈ [0, 1]. (2.1) The concepts of geometrically symmetric and strongly GA-convex functions established by Obeidat and Latif [128] and Maden et al. [66], respectively. Definition 2.2.3. A function ξ : K √ ⊆ R+ = (0,  ∞) → R is said to be geometrically symmetric with respect to cd if ξ cd x = ξ(x) for every x ∈ K. Definition 2.2.4. A function ξ : K ⊆ R+ = (0, ∞) → R is said to be strongly GA-convex with modulus µ > 0, if 2

ξ(xδ y 1−δ ) ≤ δξ(x)+(1−δ)ξ(y)−µδ(1−δ)ln y − ln x ,

∀x, y ∈ K, δ ∈ [0, 1].

Rashid et. al [140] gave the following concept of convex functions on time scales. Definition 2.2.5. Consider a time scale T and let ψ1 , ψ2 : (0, 1) → R be two nonnegative funcions. A function ξ : K = [c, d]T → R is said to be a (ψ1 , ψ2 )− convex function with respect to two nonnegative functions ψ1 and ψ2 if ξ((1 − δ)x + δy) ≤ ψ1 (1 − δ)ψ2 (δ)ξ(x) + ψ2 (1 − δ)ψ1 (δ)ξ(y), ∀ x, y ∈ K, δ ∈ [0, 1].

The following result is given by Martin and Thomas [21]. Definition 2.2.6. Let γk : T2 → R, k ∈ N0 be defined by γ0 (δ, δ1 ) = 1, ∀ δ, δ1 ∈ T and then recursively by γk+1 (δ, δ1 ) =



δ δ1

γk (ϑ, δ1 )∆ϑ, ∀ δ, δ1 ∈ T.

Gordji et al. [54] established the Hermite–Hadamard and Hermite– Hadamard-Fej´er inequalities for η-convex functions.

24

Integral Inequalities and Generalized Convexity

Theorem 2.2.1. Suppose that ξ : [c, d] → R is a η-convex function such that η is bounded from above on ξ([c, d]) × ξ([c, d]). Then, ξ



c+d 2



 d Mη 1 ≤ ξ(x)dx 2 d−c c ξ(c) + ξ(d) η(ξ(c), ξ(d)) + η(ξ(d), ξ(c)) + ≤ 2 4 ξ(c) + ξ(d) Mη ≤ + , 2 2

−

where Mη is upper bound of η. Theorem 2.2.2. (Hermite–Hadamard-Fej´er Left Inequality). Suppose that ξ : [c, d] → R is a η-convex function, such that η is bounded from above on ξ([c, d]) × ξ([c, d]). Also suppose that ψ : [c, d] → R+ is integrable and symmetric with respect to c+d 2 . Then, ξ



c+d 2



d c

ψ(x)dx −

1 2



d c

η(ξ(c + d − x), ξ(x))ψ(x)dx ≤



d

ξ(x)ψ(x)dx.

c

Theorem 2.2.3. (Hermite–Hadamard-Fej´er Right Inequality). Suppose that ξ : [c, d] → R is a η-convex function, such that η is bounded from above on ξ([c, d]) × ξ([c, d]). Also suppose that ψ : [c, d] → R+ is integrable and symmetric with respect to c+d 2 . Then, 

d c

ξ(x)ψ(x)dx ≤

ξ(c) + ξ(d) 2



d

ψ(x)dx c

η(ξ(c), ξ(d)) + η(ξ(d), ξ(c)) + 2(d − c)



c

d

(d − x)ψ(x)dx.

The following lemmas proposed by Yan Xi and Qi [175]. Lemma 2.2.1. Let ξ : K ⊆ R → R be a differentiable function on K 0 such that ξ  ∈ L1 [c, d], where c, d ∈ K with c < d. If β, γ ∈ R, then    d 1 c+d βξ(c) + γξ(d) 2 − β − γ + ξ − ξ(x)dx 2 2 2 d−c c     c+d d−c 1 (1 − β − δ)ξ  δc + (1 − δ) = 4 2 0   c + d + (1 − δ)d dδ. + (γ − δ)ξ  δ 2

Lemma 2.2.2. For m > 0 and 0 ≤ ρ ≤ 1, we have  1 ρm+1 + (1 − ρ)m+1 |ρ − δ|m dδ = m+1 0

Integral Inequalities for Strongly Generalized Convex Functions and



0

1

δ|ρ − δ|m dδ =

25

ρm+2 + (m + ρ + 1)(1 − ρ)m+1 . (m + 1)(m + 2)

We have proposed the following lemma. Lemma 2.2.3. For m > 0 and 0 ≤ ρ ≤ 1, we have  1 2ρ(1 − ρ)m+2 ρ2 (1 − ρ)m+1 2ρm+3 + (1 − ρ)m+3 + + . δ 2 |ρ − δ|m dδ = m+3 m+2 m+1 0 The following results are taken by Kwun et. al [90]. Lemma 2.2.4. Let ξ : K ⊆ R → R be a differentiable mapping on K 0 with ξ  ∈ L1 [c, d], where c, d ∈ K and c < d. Then 1 d−c



d c

ξ(x)dx − ξ



c+d 2



 1   (d − c)2 c+d 2  = + (1 − δ)c dδ δ ξ δ 16 2 0     1 c+d + dδ . (δ − 1)2 ξ  δd + (1 − δ) 2 0

Theorem 2.2.4. Let ξ : K ⊆ R → R be an η-convex function with ξ ∈ L1 [c, d], where c, d ∈ K with c < d. Then ξ





 d 1 η(ξ(c + d − x), ξ(x))dx 2(d − c) c  d 1 1 ≤ ξ(x)dx ≤ ξ(d) + η(ξ(c), ξ(d)). d−c c 2

c+d 2

−

The following lemmas proposed by Jiang et al. [68] and Obeidat and Latif [128], respectively. Lemma 2.2.5. If ξ (n) for n ∈ N exists and is integrable on [c, d], then  d n−1  (k − 1)(d − c)k ξ(c) + ξ(d) 1 − ξ (k) (c) ξ(x)dx − 2 d−c c 2(k + 1)! k=2 n  1 (d − c) δ n−1 (n − 2δ)ξ (n) (δc + (1 − δ)d)dδ. = 2n! 0 Lemma 2.2.6. For 0 < c < d, we have 1 1−δ 1+δ (1) 1 (c, d) = 0 |ln(c 2 d 2 )|dδ  (ln c)2 −3(ln d)2 +2(ln c)(ln d)  ,  4(ln d−ln c)  −(ln c)2 +3(ln d)2 −2(ln c)(ln d) = , 4(ln d−ln c)    (ln c)2 +5(ln d)2 +2(ln c)(ln d) , 4(ln d−ln c)

if d ≤ 1, √ if cd ≥ 1, √ if cd < 1 < d.

26

Integral Inequalities and Generalized Convexity  1 1−δ 1+δ 1−δ 1+δ (2) 2 (c, d) = 0 c 2 d 2 |ln(c 2 d 2 )|dδ  √ √ √ 2[d−d ln d− cd+ cd ln( cd)]  , if d ≤ 1,   ln d−ln c√ √ √ √ cd− cd ln( cd)] = 2[−d+d ln d+ , if cd ≥ 1, ln d−ln c√ √ √  √   2[2−d+d ln d− cd+ cd ln( cd)] , if cd < 1 < d. ln d−ln c

We have proposed the following lemma which are useful for our main results. Lemma 2.2.7. For 0 < c < d, we have (1) 3 (c, d) =

1 0

δc

1−δ 2

d

1+δ 2

|ln(c

1−δ 2

d

1+δ 2

)|dδ

√ √ √ √  4[d(−(ln d)2 +(ln d)(ln cd)+2 ln d−ln cd−2)− cd(ln cd−2)]  ,  2 (ln d−ln c)√  √ √ √ 4[d((ln d)2 −(ln d)(ln cd)−2 ln d+ln cd+2)+ cd(ln cd−2)] = , 2 √ (ln d−ln c) √ √ √ √    4[d((ln d)2 −(ln d)(ln cd)−2 ln d+ln cd+2)+ cd(2−ln cd)−2 ln cd−4] , (ln d−ln c)2

(2) 4 (c, d) =

1 0

δ2 c

1−δ 2

d

1+δ 2

|ln(c

1−δ 2

d

1+δ 2

if d ≤ 1, √ if cd ≥ 1, √ if cd < 1 < d.

)|dδ

 √ √ 8 [d((ln d)2 (− ln d + 2 ln cd + 3) − ln d((ln cd)2   (ln d−ln c)3  √ √ √ √ √    +4 ln cd + 6) + (ln cd)2 + 4 ln cd + 6) − 2 cd(3 − ln cd)], if d ≤ 1,   √ √  2 8  cd − 3) + ln d((ln cd)2  3 [d((ln d) (ln d − 2 ln  (ln d−ln √c) √ 2 √ √ √ √ = +4 ln cd + 6) − (ln cd) − 4 ln cd − 6) + 2 cd(3 − ln cd)], if cd ≥ 1,  √ √  8   (ln d−ln [d((ln d)3 − (ln cd)2 − 3(ln d)2 + 6 ln d + ln d(ln cd)2  c)3  √ √ √    −2(ln d)2 (ln cd) + 4 ln d ln cd − 4 ln cd − 6)   √ √ √ √ √ √  if cd < 1 < d. +2(ln cd)2 + 2 cd ln cd + 8 ln cd − 6 cd + 12],

(3) 5 (c, d) =

1 0

δ|ln(c

1−δ 2

d

1+δ 2

)|dδ

√ √  2[−(ln cd)3 −2(ln d)3 +3(ln cd)(ln d)2 ]  ,  3(ln d−ln c)2 √  √ 3 d)3 −3(ln cd)(ln d)2 ] = 2[(ln cd) +2(ln , 3(ln d−ln c)2 √ √    2[−(ln cd)3 +2(ln d)3 −3(ln cd)(ln d)2 ] , 3(ln d−ln c)2

if d ≤ 1, √ if cd ≥ 1, √ if cd < 1 < d.

1 1−δ 1+δ (4) 6 (c, d) = 0 δ 2 |ln(c 2 d 2 )|dδ √ √ √  2[(ln cd)4 +8(ln d)3 (ln cd)−6(ln d)2 (ln cd)2 −3(ln d)4 ]  ,  3(ln√d−ln c)3  √ √ 2[−(ln cd)4 −8(ln d)3 (ln cd)+6(ln d)2 (ln cd)2 +3(ln d)4 ] = , 3 3(ln √ √ d−ln c) √    2[(ln cd)4 −8(ln d)3 (ln cd)+6(ln d)2 (ln cd)2 +3(ln d)4 ] , 3(ln d−ln c)3

if d ≤ 1, √ if cd ≥ 1, √ if cd < 1 < d.

For the simplicity, we will use the following notations throughout the chapter: 1−δ 1+δ 1+δ 1−δ 1 (δ) = c 2 d 2 and 2 (δ) = c 2 d 2 . Obeidat and Latif [128] established the following lemma.

Integral Inequalities for Strongly Generalized Convex Functions

27

Lemma 2.2.8. Let ξ : K ⊆ R+ = (0, ∞) → R be a differentiable function → [0, ∞) be a continuous on K 0 and c, d ∈ K 0 with c < d, and let λ : [c, d] √ positive mapping and geometrically symmetric to cd. If ξ  ∈ L1 [c, d]√and ξ : K ⊆ R+ = (0, ∞) → R is geometrically symmetric with respect to cd, then   d (ln d)ξ(d) + (ln c)ξ(c) d (ln x)λ(x) (ln x)λ(x)ξ(x) dx − dx ln d + ln c x x c  c  1 1 (δ) (ln d − ln c) (ln x)λ(x) = dx 1 (δ) ln(1 (δ))ξ  (1 (δ))dδ 2(ln d + ln c) 0 x 2 (δ)    1  1 (δ) (ln x)λ(x)  dx 2 (δ) ln(2 (δ))ξ (2 (δ))dδ . − x 0 2 (δ) Followings lemmas are proposed by Tahir et al. [165] Lemma 2.2.9. Let ξ : T → R be a delta differentiable mapping and c, d ∈ T with c < d. If ξ ∆ ∈ Crd , then the following equality holds: ξ(c){1 − γ2 (1, 0)} + ξ(d)γ2 (1, 0) − d−c = 2



1 0



1

1 d−c

∆

0



d

ξ  (x)∆x

c

∆

[ξ (δc + (1 − δ)d) − ξ (δ1 c + (1 − δ1 )d)](δ − δ1 )∆δ∆δ1 .

Lemma 2.2.10. Let ξ : [c, d]T → R be a delta differentiable mapping on T0 and c, d ∈ T with c < d. If ξ ∆ ∈ Crd , then the following equality holds :    d 1 c+d − ξ  (x)∆x ξ 2 d−c c   d−c 1 1 ∆ = [ξ (δc + (1 − δ)d) − ξ ∆ (δ1 c + (1 − δ1 )d)](ψ(δ1 ) 2 0 0 − ψ(δ))∆δ∆δ1 , where ψ(x) =



x, x − 1,

x ∈ [0, 12 ] x ∈ ( 12 , 1].

Rashid et al. [140] gave the following corollary for delta differentiable function. Corollary 2.2.1. Consider a time scale T and K = [c, d]T such that c < d and c, d ∈ T. Suppose that there is a delta differentiable function ξ : K → R

28

Integral Inequalities and Generalized Convexity

on K 0 . If ξ ∆ ∈ Crd , then

 d 1 ξ(c) + ξ(d) − ξ  (x)∆x 2 d−c c  1   1 d−c ∆ ∆ = δξ (δd + (1 − δ)c)∆δ − δξ (δc + (1 − δ)d)∆δ . 2 0 0

Corollary 2.2.2. Consider a time scale T and K = [c, d]T such that c < d and c, d ∈ T. Suppose that there is a delta differentiable function ξ : K → R on K 0 . If ξ ∆ ∈ Crd , then    d 1 c+d − ξ  (x)∆x ξ 2 d−c c    1/2

= (d − c)

2.3

0

1

δξ ∆ (δd + (1 − δ)c)∆δ +

1/2

(δ − 1)ξ ∆ (δd + (1 − δ)c)∆δ .

Hermite–Hadamard Type Inequalities for Functions Whose Derivatives are Strongly η-Convex

In this section, we prove some Hermite–Hadamard-and Fej´er-type inequalities for strongly η-convex functions [153]. Theorem 2.3.1. Let ξ and ψ be nonnegative strongly η-convex functions with modulus µ1 and µ2 , respectively, and ξψ ∈ L1 [c, d], where c, d ∈ K, c < d. Then  d 1 ξ(x)ψ(x)dx ≤ P (c, d), (d − c) c where 1 P (c, d) = ξ(d)ψ(d) + [ξ(d)η(ψ(c), ψ(d)) + ψ(d)η(ξ(c), ξ(d))] 2 1 (c − d)2 (µ1 ψ(d) + µ2 ξ(d)) + η(ξ(c), ξ(d))η(ψ(c), ψ(d)) − 6 3 (c − d)2 µ1 µ 2 − (µ1 η(ψ(c), ψ(d)) + µ2 η(ξ(c), ξ(d))) + (c − d)4 . 12 30 Proof Since ξ and ψ are strongly η-convex functions with modulus µ1 and µ2 , respectively, therefore ξ(δc + (1 − δ)d) ≤ ξ(d) + δη(ξ(c), ξ(d)) − µ1 δ(1 − δ)(c − d)2 , ∀ δ ∈ [0, 1] (2.2) and ψ(δc+(1−δ)d) ≤ ψ(d)+δη(ψ(c), ψ(d))−µ2 δ(1−δ)(c−d)2 , ∀ δ ∈ [0, 1]. (2.3)

Integral Inequalities for Strongly Generalized Convex Functions

29

From (2.2) and (2.3), we obtain ξ(δc + (1 − δ)d)ψ(δc + (1 − δ)d) ≤ ξ(d)ψ(d) + δ(ξ(d)η(ψ(c), ψ(d)) + ψ(d)η(ξ(c), ξ(d))) + δ 2 η(ξ(c), ξ(d))η(ψ(c), ψ(d))

− δ 2 (1 − δ)(c − d)2 (µ1 η(ψ(c), ψ(d)) + µ2 η(ξ(c), ξ(d)))

− δ(1 − δ)(c − d)2 (µ1 ψ(d) + µ2 ξ(d)) + δ 2 (1 − δ)2 µ1 µ2 (c − d)4 . Integrating above inequality from 0 to 1 on both sides with respect to δ, we have  1 ξ(δc + (1 − δ)d)ψ(δc + (1 − δ)d)dδ ≤ ξ(d)ψ(d) 0

1 1 + [ξ(d)η(ψ(c), ψ(d)) + ψ(d)η(ξ(c), ξ(d))] + η(ξ(c), ξ(d))η(ψ(c), ψ(d)) 2 3 (c − d)2 (c − d)2 (µ1 η(ψ(c), ψ(d)) + µ2 η(ξ(c), ξ(d))) − (µ1 ψ(d) − 12 6 µ1 µ 2 (c − d)4 . + µ2 ξ(d)) + 30

This implies, 1 (d − c)



c

d

ξ(x)ψ(x)dx ≤ P (c, d).

This completes the proof. Remark 2.3.1. When µ1 = 0 and µ2 = 0, then above theorem reduces to Theorem 2.2 of [90]: i.e.  d 1 ξ(x)ψ(x)dx ≤ P1 (c, d), (d − c) c where 1 P1 (c, d) = ξ(d)ψ(d) + [ξ(d)η(ψ(c), ψ(d)) + ψ(d)η(ξ(c), ξ(d))] 2 1 + η(ξ(c), ξ(d))η(ψ(c), ψ(d)). 3 Remark 2.3.2. If µ = 0 and η(x, y) = x − y, then above theorem reduces to Theorem 1 of [130]: i.e.  d 1 ξ(x)ψ(x)dx ≤ P2 (c, d), (d − c) c where 1 P2 (c, d) = ξ(d)ψ(d) + [ξ(d)(ψ(c) − ψ(d)) + ψ(d)(ξ(c) − ξ(d))] 2 1 + (ξ(c) − ξ(d))(ψ(c) − ψ(d)). 3

30

Integral Inequalities and Generalized Convexity

Theorem 2.3.2. Let ξ be an strongly η-convex function with modulus µ and ξ ∈ L1 [c, d], where c, d∈ K, c < d and ψ : [c, d] → R be nonnegative, integrable and symmetric about c+d . Then 2 

d

c

 d  1 ξ(x)ψ(x)dx ≤ ξ(d) + η(ξ(c), ξ(d)) ψ(x)dx 2 c  d −µ (x − c)(d − x)ψ(x)dx. c

Proof Since, ξ be an strongly η-convex function  with modulus µ, and ψ  , therefore we have nonnegative, integrable and symmetric about c+d 2 

d

c

ξ(x)ψ(x)dx    d d 1 = ξ(x)ψ(x)dx + ξ(c + d − x)ψ(c + d − x)dx 2 c c    d d 1 = ξ(x)ψ(x)dx + ξ(c + d − x)ψ(x)dx 2 c c       1 d x−c x−c d−x d−x = c+ d +ξ c+ d ψ(x)dx ξ 2 c d−c d−c d−c d−c     1 d d−x ≤ η(ξ(c), ξ(d)) ξ(d) + 2 c d−c        x−c x−c d−x 2 − µ (c − d) + ξ(d) + η(ξ(c), ξ(d)) d−c d−c d−c     d−x x−c (c − d)2 ψ(x)dx − µ d−c d−c  d   d 1 = ξ(d) + η(ξ(c), ξ(d)) ψ(x)dx − µ (x − c)(d − x)ψ(x)dx. 2 c c

This completes the proof. Remark 2.3.3. When µ = 0, then above theorem reduces to Theorem 2.3 of [90]: i.e 

 d  1 ξ(d) + η(ξ(c), ξ(d)) ψ(x)dx. 2 c

d c

ξ(x)ψ(x)dx ≤

Remark 2.3.4. If µ = 0, η(x, y) = x − y and ψ(x) = 1, then above theorem reduces to the second inequality of Theorem 1.3.1, i.e 1 (d − c)



c

d

ξ(x)dx ≤

1 (ξ(c) + ξ(d)). 2

Integral Inequalities for Strongly Generalized Convex Functions

31

Now we establish the results on integral inequalities for strongly η-convex functions. Theorem 2.3.3. Let ξ : K ⊆ R → R, be a differentiable mapping on K 0 with ξ  ∈ L1 [c, d], where c, d ∈ K, c < d. If |ξ  (x)|q for q ≥ 1 is strongly η-convex with modulus µ on [c, d] and 0 ≤ β, γ ≤ 1, then    d  βξ(c) + γξ(d) 2 − β − γ  c + d   1   + ξ − ξ(x)dx    2 2 2 d−c c

≤



d−c 8



1 24

1/q 

1

(2β 2 − 2β + 1)1− q (24(2β 2 − 2β + 1)|ξ  (d)|q

+ 4(−2β 3 + 12β 2 − 9β + 4)η(|ξ  (c)|q , |ξ  (d)|q )

− µ(−7β 4 + 28β 3 − 30β 2 + 12β)(c − d)2 )1/q 1

+ (2γ 2 − 2γ + 1)1− q (24(2γ 2 − 2γ + 1)|ξ  (d)|q + 4(2γ 3 − 3γ + 2)η(|ξ  (c)|q , |ξ  (d)|q )

 − µ(−7γ 4 + 8γ 3 − 8γ + 5)(c − d)2 )1/q .

Proof Recall Lemma 2.2.1;

   d 1 βξ(c) + γξ(d) 2 − β − γ c+d + ξ − ξ(x)dx 2 2 2 d−c c     c+d d−c 1 (1 − β − δ)ξ  δc + (1 − δ) = 4 2 0   c+d + (1 − δ)d dδ. + (γ − δ)ξ  δ 2 This implies,    d  βξ(c) + γξ(d) 2 − β − γ  c + d   1   + ξ − ξ(x)dx    2 2 2 d−c c    1   d−c c + d  ≤ |1 − β − δ| ξ  δc + (1 − δ)  dδ 4 2 0      1   c+d + + (1 − δ)d  dδ . |γ − δ| ξ  δ 2 0

(2.4)

32

Integral Inequalities and Generalized Convexity

Applying H¨ older’s inequality and the definition of strong η-convexity in (2.4), we have    d  βξ(c) + γξ(d) 2 − β − γ  c + d   1   + ξ − ξ(x)dx    2 2 2 d−c c  1 1− q  1 1 d−c |1 − β − δ|dδ |1 − β − δ| (|ξ  (d)|q ≤ 4 0 0  1/q      1−δ 1+δ 1+δ η(|ξ  (c)|q , |ξ  (d)|q ) − µ (c − d)2 dδ + 2 2 2  1 1− q1  1    δ η(|ξ  (c)|q , |ξ  (d)|q ) + |γ − δ|dδ |γ − δ| |ξ  (d)|q + 2 0 0    1/q   δ δ 2 − µ 1− (c − d) dδ . (2.5) 2 2 Using Lemmas 2.2.2 and 2.2.3, we calculate      1 µ 1+δ η(|ξ  (c)|q , |ξ  (d)|q ) − (1 − δ 2 )(c − d)2 dδ |1 − β − δ| |ξ  (d)|q + 2 4 0   1 1 µ = |ξ  (d)|q + η(|ξ  (c)|q , |ξ  (d)|q ) − (c − d)2 |1 − β − δ|dδ 2 4 0  1  1 1 µ  q  q 2 + η(|ξ (c)| , |ξ (d)| ) δ|1 − β − δ|dδ + (c − d) δ 2 |1 − β − δ|dδ 2 4 0 0 1 1 = (2β 2 − 2β + 1)|ξ  (d)|q + (−2β 3 + 12β 2 − 9β + 4)η(|ξ  (c)|q , |ξ  (d)|q ) 2 12 µ 4 3 2 − (−7β + 28β − 30β + 12β)(c − d)2 (2.6) 48 and  1

       δ δ δ η(|ξ  (c)|q , |ξ  (d)|q ) − µ 1− (c − d)2 dδ |γ − δ| |ξ  (d)|q + 2 2 2 0   1  1 1 µ = |ξ  (d)|q η(|ξ  (c)|q , |ξ  (d)|q ) − (c − d)2 |γ − δ|dδ + δ|γ − δ|dδ 2 2 0 0  1 µ + (c − d)2 δ 2 |γ − δ|dδ 4 0 1 1 = (2γ 2 − 2γ + 1)|ξ  (d)|q + (2γ 3 − 3γ + 2)η(|ξ  (c)|q , |ξ  (d)|q ) 2 12 µ − (−7γ 4 + 8γ 3 − 8γ + 5)(c − d)2 . (2.7) 48

Integral Inequalities for Strongly Generalized Convex Functions

33

From (2.5)–(2.7) and Lemma 2.2.2, we have    d  βξ(c) + γξ(d) 2 − β − γ  c + d   1   + ξ − ξ(x)dx    2 2 2 d−c c ≤



d−c 8



1 24

1/q 

1

(2β 2 − 2β + 1)1− q (24(2β 2 − 2β + 1)|ξ  (d)|q

+ 4(−2β 3 + 12β 2 − 9β + 4)η(|ξ  (c)|q , |ξ  (d)|q ) − µ(−7β 4 + 28β 3 − 30β 2 + 12β)(c − d)2 )1/q 1

+ (2γ 2 − 2γ + 1)1− q (24(2γ 2 − 2γ + 1)|ξ  (d)|q + 4(2γ 3 − 3γ + 2)η(|ξ  (c)|q , |ξ  (d)|q )

 −µ(−7γ 4 + 8γ 3 − 8γ + 5)(c − d)2 )1/q .

This completes the proof.

Corollary 2.3.1. If β = γ in above theorem, then      d β  c+d 1   − ξ(x)dx  (ξ(c) + ξ(d)) + (1 − β)ξ 2  2 d−c c ≤



d−c 8



1 24

1/q

1

(2β 2 − 2β + 1)1− q

× [(24(2β 2 − 2β + 1)|ξ  (d)|q + 4(−2β 3 + 12β 2 − 9β + 4)η(|ξ  (c)|q , |ξ  (d)|q ) − µ(−7β 4 + 28β 3 − 30β 2 + 12β)(c − d)2 )1/q + (24(2β 2 − 2β + 1)|ξ  (d)|q + 4(2β 3 − 3β + 2)η(|ξ  (c)|q , |ξ  (d)|q ) − µ(−7β 4 + 8β 3 − 8β + 5)(c − d)2 )1/q ]. Corollary 2.3.2. If β = γ = 12 in Corollary 2.3.1, then       d   1 ξ(c) + ξ(d) c+d 1   +ξ − ξ(x)dx   2 2 2 d−c c ≤



d−c 16



1 192

1/q

[(192|ξ  (d)|q + 144η(|ξ  (c)|q , |ξ  (d)|q ) − 25µ(c − d)2 )1/q

+ (192|ξ  (d)|q + 48η(|ξ  (c)|q , |ξ  (d)|q ) − 25µ(c − d)2 )1/q ].

34

Integral Inequalities and Generalized Convexity

Corollary 2.3.3. If q = 1 in Corollary 2.3.2, then       d  1 ξ(c) + ξ(d)  c+d 1   +ξ − ξ(x)dx  2  2 2 d−c c   d−c [192|ξ  (d)| + 96η(|ξ  (c)|, |ξ  (d)|) − 25µ(c − d)2 ]. ≤ 1536 Remark 2.3.5. When µ = 0, then above theorem reduces to Theorem 3.1 of [90]: i.e.    d  βξ(c) + γξ(d) 2 − β − γ  c + d   1   + ξ − ξ(x)dx    2 2 2 d−c c   1/q   1 1 d−c (2β 2 − 2β + 1)1− q (24(2β 2 − 2β + 1)|ξ  (d)|q ≤ 8 24 + 4(−2β 3 + 12β 2 − 9β + 4)η(|ξ  (c)|q , |ξ  (d)|q ) 1

+ (2γ 2 − 2γ + 1)1− q (24(2γ 2 − 2γ + 1)|ξ  (d)|q  + 4(2γ 3 − 3γ + 2)η(|ξ  (c)|q , |ξ  (d)|q ) .

Theorem 2.3.4. Let ξ : K ⊆ R → R, be a differentiable mapping on K 0 with ξ  ∈ L1 [c, d], where c, d ∈ K, c < d. If |ξ  (x)|q for q ≥ 1 is strongly η-convex with modulus µ on [c, d] and 0 ≤ β, γ ≤ 1, then    d  βξ(c) + γξ(d) 2 − β − γ  c + d   1   + ξ − ξ(x)dx    2 2 2 d−c c   d−c (1 − β)q+1 + β q+1 |ξ  (d)|q ≤ 4 q+1   (q + 2)((1 − β)q+1 + 2β q+1 ) + (1 − β)q+2 − β q+2 η(|ξ  (c)|q , |ξ  (d)|q ) + 2(q + 1)(q + 2)  2(1 − β)β q+2 (1 − β)q+1 − β q+3 + 2β q+2 µ 2 − − (c − d) 4 q+1 q+2    1/q 2(1 − β)q+3 β q+3 (1 − γ)q+1 + γ q+1 − |ξ  (d)|q + q+3 q+1   (q + γ + 1)(1 − γ)q+1 + γ q+2 η(|ξ  (c)|q , |ξ  (d)|q ) + 2(q + 1)(q + 2)  µ −2(q + 1)γ(1 − γ)q+2 − (q + 2)γ 2 (1 − γ)q+1 + 2γ q+2 − (c − d)2 4 (q + 1)(q + 2) 1/q  2(q + γ + 1)(1 − γ)q+1 2γ q+3 + (1 − γ)q+3 + − . (q + 1)(q + 2) q+3

Integral Inequalities for Strongly Generalized Convex Functions Proof From Lemma 2.2.1, we have    d  βξ(c) + γξ(d) 2 − β − γ  c + d   1   + ξ − ξ(x)dx    2 2 2 d−c c    1   d−c c + d  ≤ |1 − β − δ| ξ  δc + (1 − δ)  dδ 4 2 0      1   c+d + + (1 − δ)d  dδ . |γ − δ| ξ  δ 2 0

35

Using H¨ older’s inequality and the definition of strong η-convexity, we have    d  βξ(c) + γξ(d) 2 − β − γ  c + d   1   + ξ − ξ(x)dx    2 2 2 d−c c  1   q 1/q 1− q  1 1  c + d  d−c dδ |1 − β − δ|q ξ  δc + (1 − δ) ≤  dδ 4 2 0 0   q 1/q   1 1− q1  1   c+d q  + (1 − δ)d  dδ + dδ |γ − δ| ξ δ 2 0 0    1+δ η(|ξ  (c)|q , |ξ  (d)|q ) |1 − β − δ|q |ξ  (d)|q + 2 0  1/q  1 µ 2 2 − (1 − δ )(c − d) dδ + |γ − δ|q (|ξ  (d)|q 4 0       1/q  δ δ δ  q  q 2 η(|ξ (c)| , |ξ (d)| ) − µ 1− (c − d) dδ + 2 2 2   d−c 1 µ = |ξ  (d)|q + η(|ξ  (c)|q , |ξ  (d)|q ) − (c − d)2 4 2 4  1  1 1 × |1 − β − δ|q dδ + η(|ξ  (c)|q , |ξ  (d)|q ) δ|1 − β − δ|q dδ 2 0 0 1/q  1 µ δ 2 |1 − β − δ|q dδ + (c − d)2 4 0     1 µ 1 η(|ξ  (c)|q , |ξ  (d)|q ) − (c − d)2 + |ξ  (d)|q |γ − δ|q dδ + 2 2 0 1/q   1  1 µ . δ|γ − δ|q dδ + (c − d)2 δ 2 |γ − δ|q dδ × 4 0 0 ≤

d−c 4



1

36

Integral Inequalities and Generalized Convexity

Applying Lemmas 2.2.2 and 2.2.3, we obtain    d  βξ(c) + γξ(d) 2 − β − γ  c + d   1   + ξ − ξ(x)dx    2 2 2 d−c c   q+1 q+1 (1 − β) d−c +β |ξ  (d)|q ≤ 4 q+1   (q + 2)((1 − β)q+1 + 2β q+1 ) + (1 − β)q+2 − β q+2 + 2(q + 1)(q + 2)  µ (1 − β)q+1 − β q+3 + 2β q+2  q  q 2 × η(|ξ (c)| , |ξ (d)| ) − (c − d) 4 q+1 1/q  q+2 q+3 q+3 2(1 − β)β 2(1 − β) +β − − q+2 q+3   d−c (1 − γ)q+1 + γ q+1 |ξ  (d)|q + 4 q+1   (q + γ + 1)(1 − γ)q+1 + γ q+2 µ η(|ξ  (c)|q , |ξ  (d)|q ) − (c − d)2 + 2(q + 1)(q + 2) 4  q+2 2 q+1 q+2 −2(q + 1)γ(1 − γ) − (q + 2)γ (1 − γ) + 2γ × (q + 1)(q + 2) 1/q  2(q + γ + 1)(1 − γ)q+1 2γ q+3 + (1 − γ)q+3 + . − (q + 1)(q + 2) q+3 This completes the proof. Corollary 2.3.4. If β = γ in then    abovetheorem,   β d 1 c+d   − ξ(x)dx  (ξ(c) + ξ(d)) + (1 − β)ξ  2 2 d−c c   d−c (1 − β)q+1 + β q+1 |ξ  (d)|q ≤ 4 q+1   (q + 2)((1 − β)q+1 + 2β q+1 ) + (1 − β)q+2 − β q+2 η(|ξ  (c)|q , |ξ  (d)|q ) + 2(q + 1)(q + 2)  2(1 − β)β q+2 µ (1 − β)q+1 − β q+3 + 2β q+2 − − (c − d)2 4 q+1 q+2 1/q   q+3 q+3 q+1 q+1 2(1 − β) β (1 − β) +β − |ξ  (d)|q + q+3 q+1   (q + β + 1)(1 − β)q+1 + β q+2 η(|ξ  (c)|q , |ξ  (d)|q ) + 2(q + 1)(q + 2)  −2(q + 1)β(1 − β)q+2 − (q + 2)β 2 (1 − β)q+1 + 2β q+2 µ − (c − d)2 4 (q + 1)(q + 2) 1/q  q+1 q+3 2(q + β + 1)(1 − β) 2β + (1 − β)q+3 + − . (q + 1)(q + 2) q+3

Integral Inequalities for Strongly Generalized Convex Functions

37

Remark 2.3.6. When µ = 0, then above theorem reduces to Theorem 3.2 of [90], i.e.    d  βξ(c) + γξ(d) 2 − β − γ  c + d   1   + ξ − ξ(x)dx    2 2 2 d−c c   (1 − β)q+1 + β q+1 d−c |ξ  (d)|q ≤ 4 q+1   1/q (q + 2)((1 − β)q+1 + 2β q+1 ) + (1 − β)q+2 − β q+2  q  q η(|ξ (c)| , |ξ (d)| ) + 2(q + 1)(q + 2)     (1 − γ)q+1 + γ q+1 (q + γ + 1)(1 − γ)q+1 + γ q+2  q |ξ (d)| + + q+1 2(q + 1)(q + 2)   q  q 1/q × η(|ξ (c)| , |ξ (d)| )) . Theorem 2.3.5. Let ξ : K ⊂ [0, ∞) → R be a differentiable mapping on K 0 with ξ  ∈ L1 [c, d], where c, d ∈ K and c < d. If |ξ  | is strongly η-convex with modulus µ on [c, d], then      d   c+d 1   − ξ(x)dx ξ   2 d−c c           2   c + d     c + d  1 (d − c) 1      |ξ (c)| + ξ ≤  + 4 η ξ  , |ξ (c)| 16 3 2 2       1 µ c + d  − (d − c)2 . + η |ξ  (d)|, ξ   3 2 40

Proof Recall Lemma 2.2.4, we have

     d   1 c+d   − ξ(x)dx ξ   2 d−c c     1   (d − c)2 c+d 2   + (1 − δ)c  dδ ≤ δ ξ δ 16 2 0      1  c + d  + (δ − 1)2 ξ  δd + (1 − δ)  dδ . 2 0

38

Integral Inequalities and Generalized Convexity

Using the definition of strong η-convexity, we obtain      d   c+d 1   − ξ(x)dx ξ   2 d−c c       1   c + d   (d − c)2 2    ≤ δ |ξ (c)| + δη ξ  , |ξ (c)| 16 2 0   2 d−c dδ − µδ(1 − δ) 2         1   c + d   c + d    ξ + + δη |ξ (δ − 1)2 ξ  (d)|,    2 2 0   2  d−c − µδ(1 − δ) dδ 2      2     c + d   (d − c)2 1 µ 1  d − c  , |ξ (c)| − |ξ (c)| + η ξ =  16 3 4 2 20 2          2    c + d  1   c + d  1 µ d−c    + ξ  + 12 η |ξ (d)|, ξ  − 20 3 2 2 2           2   c + d     c + d  1 (d − c) 1      |ξ (c)| + ξ =  + 4 η ξ  , |ξ (c)| 16 3 2 2       µ c + d  1 − (d − c)2 . + η |ξ  (d)|, ξ   3 2 40

This completes the proof.

Remark 2.3.7. When µ = 0, then above theorem reduces to Theorem 3.3 of [90]: i.e.      d   c+d 1   − ξ(x)dx ξ   2 d−c c             c + d    1 c + d  (d − c)2 1 ξ  , |ξ (c)| |ξ  (c)| + ξ  + η ≤    16 3 2 4 2      1 c + d  + η |ξ  (d)|, ξ  .  3 2

Theorem 2.3.6. Let ξ : K ⊂ [0, ∞) → R be a differentiable mapping on K 0 with ξ  ∈ L1 [c, d], where c, d ∈ K and c < d. If |ξ  |q for q ≥ 1 with p1 + 1q = 1

Integral Inequalities for Strongly Generalized Convex Functions is strongly η-convex with modulus µ on [c, d], then      d   c+d 1   − ξ(x)dx ξ   2 d−c c 1          c + d q  1 1  (d − c)2 1 p q q   |ξ (c)| + η ξ ≤  , |ξ (c)| 16 3 3 4 2   q1  1   c + d q µ ξ   − (d − c)2 +  80 3 2    q   q1   1 c + d  µ  q   2 η |ξ (d)| , ξ + .  − 80 (d − c) 12 2 Proof From Lemma 2.2.4, we have      d   1 c+d   − ξ(x)dx ξ   2 d−c c     1   c+d (d − c)2 2   + (1 − δ)c  dδ ≤ δ ξ δ 16 2 0      1  c + d  2   + (δ − 1) ξ δd + (1 − δ)  dδ . 2 0

Using H¨ older’s inequality, we have

     d   1 c+d   − ξ(x)dx ξ   2 d−c c     1  2 2  (d − c)2 c+d + (1 − δ)c  dδ ≤ δ p δ q ξ  δ 16 2 0      1  2 2 c + d    p q + (δ − 1) (δ − 1) ξ δd + (1 − δ)  dδ 2 0  q  q1  p1  1   1   c+d (d − c)2 2 2   + (1 − δ)c  dδ δ dδ δ ξ δ ≤ 16 2 0 0   q  q1   1  p1  1   c + d  dδ + (δ − 1)2 dδ (δ − 1)2 ξ  δd + (1 − δ) .  2 0 0

39

40

Integral Inequalities and Generalized Convexity

Since |ξ  | is strongly η-convex function with modulus µ > 0, therefore      d   1 c+d   − ξ(x)dx ξ   2 d−c c  1 q  p  1     1  c + d  (d − c)2  q δ 2 dδ δ 2 |ξ  (c)|q + δη ξ  , |ξ (c)| ≤  16 2 0 0   q  p1  1   q1  1  µ c + d  2 2 2   (δ − 1) dδ (δ − 1) ξ − δ(1 − δ)(d − c) dδ +  4 2 0 0   q    q1    µ c + d  − δ(1 − δ)(d − c)2 dδ + δη |ξ  (d)|q , ξ   2 4   1    q   1 p 1  1 c + d  (d − c)2  q |ξ (c)|q + η ξ  , |ξ (c)| =  16 3 3 4 2  1  1  p1  1   c + d q µ 2 q ξ   (d − c) + −  80 3 3 2    q   q1   1 c + d  µ  q   2 η |ξ (d)| , ξ + .  − 80 (d − c) 12 2 This completes the proof.

Remark 2.3.8. When µ = 0, then above theorem reduces to Theorem 3.4 of [90]: i.e.      d   1 c+d   − ξ(x)dx ξ   2 d−c c     1    q1   c + d q  1 1  (d − c)2 1 p q q  , |ξ (c)| |ξ (c)| + η ξ ≤  16 3 3 4 2    q q  q1      1 c + d  1   c + d   q   ξ + .  + 12 η |ξ (d)| , ξ  3 2 2

2.3.1

Application to special means

Now, we consider the following special means for positive real numbers c, d > 0 [153]: c+d . Arithmetic mean: A(c, d) = 2 √ Geometric mean: G(c, d) = cd. Harmonic mean :

H(c, d) =

2

1 1 c+d

.

Integral Inequalities for Strongly Generalized Convex Functions Generalized logarithmic mean:    dm+1 −cm+1 1/m , (m+1)(d−c) L(c, d) = c,

if c = d, m = −1, 0 if c = d.

    1 dd 1/(d−c)

, if d = c, e cc c, if c = d.   cm +w(cd) m2 +dm 1/m , if m = 0, (w+2) Hw,m (c, d) = √  cd, if m = 0,

Identric mean:

Heronian mean:

41

I(c, d) =

for 0 ≤ w < ∞. Now, using the above results in previous theorems, we have some applications to the special means of positive real numbers. Theorem 2.3.7. Let c, d > 0, c = d, q ≥ 1, and either m > 1 and (m−1)q ≥ 1 or m < 0. Then     A(βcm , γdm ) + 2 − β − γ Am (c, d) − Lm (c, d)   2   1/q   1 1 d−c (2β 2 − 2β + 1)1− q (24(2β 2 − 2β + 1)|mdm−1 |q ≤ 8 24 + 4(−2β 3 + 12β 2 − 9β + 4)η(|mcm−1 |q , |mdm−1 )|q )

1

− µ(−7β 4 + 28β 3 − 30β 2 + 12β)(c − d)2 )1/q + (2γ 2 − 2γ + 1)1− q

× (24(2γ 2 − 2γ + 1)|mdm−1 )|q + 4(2γ 3 − 3γ + 2)η(|mcm−1 |q , |mdm−1 |q )  − µ(−7γ 4 + 8γ 3 − 8γ + 5)(c − d)2 )1/q . Proof Applying Theorem 2.3.3 with ξ(x) = xm . Then we obtain the result immediately. Example 2.3.1. Let ξ(x) = x2 , η(x, y) = x + y + (x − y)2 , µ = 1, β = γ = 1, c = 1, d = 2, q = 1. Then above theorem is verified.

42

Integral Inequalities and Generalized Convexity

Theorem 2.3.8. Let c, d > 0, c = d, q ≥ 1, Then     ln G2 (cβ , dγ ) 2 − β − γ   + ln A(c, d) − ln I(c, d)   2 2   1/q    q  1 1 d−c 1− q1 2 2 (2β − 2β + 1) 24(2β − 2β + 1) ≤ 8 24 d  q  q  1 1 + 4(−2β 3 + 12β 2 − 9β + 4)η , c d  1 4 3 2 2 1/q + (2γ 2 − 2γ + 1)1− q − µ(−7β + 28β − 30β + 12β)(c − d)   q  q  q  1 1 1 2 3 × 24(2γ − 2γ + 1) + 4(2γ − 3γ + 2)η , d c d 1/q  − µ(−7γ 4 + 8γ 3 − 8γ + 5)(c − d)2 .

Proof Applying Theorem 2.3.3 with ξ(x) = ln x. Then we obtain the result immediately. Theorem 2.3.9. For d > c > 0, c = dw ≥ 0, and m ≥ 4 or 0 = m < 1, we have   m        1 Hw,m (c, d)  d c c d m m    2 H(cm , dm ) + Hw,m d + c , 1 − Hw,m L d , c , 1        (d − c)A(c, d) 192|m| 1 1 w 2( m −1) 2(m−1) 2 ≤ d, d, G + G 768 G2 (c, d) w + 2 c 2 c       m w |m| 1 1 G2(m−1) c, + G2( 2 −1) c, , + 96η w+2 d 2 d       w 2( m −1) 100µ(d − c)2 A2 (c, d) 1 1 |m| 2(m−1) 2 G + G − . d, d, w+2 c 2 c G2 (c2 , d2 )

Proof From Corollary 2.3.3, we have         dc  1 ξ c + ξ d c d 1   d c d + c +ξ − d c ξ(x)dx  c  2 2 2 − c d d            2    c   d c    − d d d c    + 96η ξ   − 25µ ≤ c d . 192 ξ  −  , ξ  1536 c  d c  d c

(2.8)

Integral Inequalities for Strongly Generalized Convex Functions

43

m xm +wx 2

+1 Applying ξ(x) = for x > 0 and m ∈ / (1, 4) in above inequality, we w+2 obtain     ξ dc + ξ dc 2  d m  c  m2  d  m2  c m + w + 1 + w +1 d c + c (2.9) = d 2(w + 2) 2(w + 2)   2m m m 1 c + wcm (cd) 2 + 2cm dm + +wdm (cd) 2 + d2m = 2(w + 2) c m dm   m m m 1 (c + w(cd) 2 + d )(cm + dm ) = 2(w + 2) c m dm m Hw,m (c, d) = , (2.10) H(cm , dm )

ξ



d c

and 

c d

1 −

+ 2

c d

d c



 d c

c

d d+c

=

2

m

+w

c

d d+c

2

(w + 2)

 m2

+1

m = Hw,m



c d

 + dc ,1 , 2

(2.11)

ξ(x)dx

c d

     m m ( dc ) 2 +1 − ( dc ) 2 +1 ( dc )m+1 − ( dc )m+1 1 +w +1 = d c (w + 2) (m + 1)( dc − dc ) (m 2 + 1)( c − d )     c d m = Hw,m , ,1 , (2.12) L d c 

          2    d    d  c d c     + 96η ξ  − 25µ − 192 ξ  , ξ c  d c  d c       m (d − c)A(c, d) 192|m| 1 1 w = G2(m−1) d, + G2( 2 −1) d, 2 768 G (c, d) w + 2 c 2 c       m w |m| 1 1 G2(m−1) c, + G2( 2 −1) c, , + 96η w+2 d 2 d      m w 1 1 |m| G2(m−1) d, + G2( 2 −1) d, w+2 c 2 c  100µ(d − c)2 A2 (c, d) − . (2.13) G2 (c2 , d2 ) d c

− dc 1536

44

Integral Inequalities and Generalized Convexity

Applying (2.9) and (2.11)–(2.13) in (2.8), we have   m        1 Hw,m (c, d)  d c c d m m  + H + , 1 − H , , 1  L w,m w,m  2 H(cm , dm ) d c d c       (d − c)A(c, d) 192|m| 1 1 w 2( m −1) 2(m−1) 2 ≤ d, d, G + G 768 G2 (c, d) w + 2 c 2 c       m w |m| 1 1 G2(m−1) c, + G2( 2 −1) c, , + 96η w+2 d 2 d      m w 1 1 |m| G2(m−1) d, + G2( 2 −1) d, w+2 c 2 c  2 2 100µ(d − c) A (c, d) − . G2 (c2 , d2 )

2.4

Weighted Version of Hermite–Hadamard Type Inequalities for Strongly GA-Convex Functions

We shall establish some weighted Hermite–Hadamard inequalities for strongly GA-convex functions by using geometric symmetry of a continuous positive mapping and a differentiable mapping whose derivatives in absolute value are strongly GA-convex [159]. Theorem 2.4.1. Let ξ : K ⊆ R+ = (0, ∞) → R be a differentiable function on K 0 and c, d ∈ K 0 with c < d, and let λ : [c, d] √ → [0, ∞) be1 a continuous [c, d], ξ : K ⊆ positive mapping and geometrically symmetric to cd. If ξ  ∈ L√ R+ = (0, ∞) → R is geometrically symmetric with respect to cd and |ξ  | is strongly GA-convex on [c, d] with modulus µ > 0, then    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c   2 (ln d − ln c) |ξ (c)| λ∞ (∆2 (c, d) + ∆2 (d, c) − ∆3 (c, d) + ∆3 (d, c)) ≤ 8 2 |ξ  (d)| + (∆2 (c, d) + ∆2 (d, c) + ∆3 (c, d) − ∆3 (d, c)) 2  µ − ln d − ln c2 (∆2 (c, d) + ∆2 (d, c) − ∆4 (c, d) − ∆4 (d, c)) , 4 where λ∞ = supx∈[c,d] |λ(x)|.

Integral Inequalities for Strongly Generalized Convex Functions

45

Proof For the proof of this theorem, we will use Lemma 2.2.8.   d (ln d)ξ(d) + (ln c)ξ(c) d (ln x)λ(x) (ln x)λ(x)ξ(x) dx − dx ln d + ln c x x c  c  1 1 (δ) (ln d − ln c) (ln x)λ(x) = dx 1 (δ) ln(1 (δ))ξ  (1 (δ))dδ 2(ln d + ln c) 0 x 2 (δ)    1  1 (δ) (ln x)λ(x) dx 2 (δ) ln(2 (δ))ξ  (2 (δ))dδ . − x 0 2 (δ) This implies,    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c    1 1 (δ) (ln d − ln c) ln x ≤ λ∞ dx 1 (δ)|ln(1 (δ))||ξ  (1 (δ))|dδ 2(ln d + ln c) x 0 2 (δ)    1  1 (δ) ln x  dx 2 (δ)|ln(2 (δ))||ξ (2 (δ))|dδ . + (2.14) x 0 2 (δ) Since |ξ  | is strongly GA-convex function on [c, d] with modulus µ > 0, we have |ξ  (1 (δ))| = |ξ  (c ≤

1−δ 2

d

1+δ 2

)|

(1 − δ)  (1 + δ)  µ 2 |ξ (c)| + |ξ (d)| − (1 − δ)(1 + δ)ln d − ln c 2 2 4 (2.15)

and |ξ  (2 (δ))| = |ξ  (c ≤

1+δ 2

d

1−δ 2

)|

(1 + δ)  (1 − δ)  µ 2 |ξ (c)| + |ξ (d)| − (1 + δ)(1 − δ)ln d − ln c . 2 2 4 (2.16)

Using (2.15) and (2.16) in (2.14), we have    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c   1 (ln d − ln c)2 (1 + δ)  (1 − δ)  ≤ λ∞ |ξ (c)| + |ξ (d)| 8 2 2 0  1  µ (1 + δ)  2 2 − (1 − δ )ln d − ln c 1 (δ)|ln(1 (δ))|dδ + |ξ (c)| 4 2 0   µ (1 − δ)  2 2 |ξ (d)| − (1 − δ )ln d − ln c 2 (δ)|ln(2 (δ))|dδ . + 2 4

46

Integral Inequalities and Generalized Convexity

By applying Lemmas 2.2.6 and 2.2.7, we have    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c  (ln d − ln c)2 |ξ  (c)| λ∞ (∆2 (c, d) + ∆2 (d, c) − ∆3 (c, d) + ∆3 (d, c)) ≤ 8 2 |ξ  (d)| + (∆2 (c, d) + ∆2 (d, c) + ∆3 (c, d) − ∆3 (d, c)) 2  µ 2 − ln d − ln c (∆2 (c, d) + ∆2 (d, c) − ∆4 (c, d) − ∆4 (d, c)) . 4 This completes the proof.

Corollary 2.4.1. If λ(x) = (ln x)(ln1 d−ln c) , ∀ x ∈ [c, d] with 1 < c < d < ∞ in Theorem 2.4.1, then    d  (ln d)ξ(d) + (ln c)ξ(c) 1 ξ(x)   − dx    ln d + ln c ln d − ln c c x   (ln d − ln c) |ξ (c)| ≤ (∆2 (c, d) + ∆2 (d, c) − ∆3 (c, d) + ∆3 (d, c)) 8(ln c) 2 |ξ  (d)| + (∆2 (c, d) + ∆2 (d, c) + ∆3 (c, d) − ∆3 (d, c)) 2  µ 2 − ln d − ln c (∆2 (c, d) + ∆2 (d, c) − ∆4 (c, d) − ∆4 (d, c)) . 4

Corollary 2.4.2. If µ = 0 in Theorem 2.4.1, then    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c  (ln d − ln c)2 |ξ  (c)| λ∞ (∆2 (c, d) + ∆2 (d, c) − ∆3 (c, d) + ∆3 (d, c)) ≤ 8 2  |ξ  (d)| (∆2 (c, d) + ∆2 (d, c) + ∆3 (c, d) − ∆3 (d, c)) , + 2 where λ∞ = supx∈[c,d] |λ(x)|.

Remark 2.4.1. If |ξ  | is geometrically quasi convex, then above theorem reduces to Theorem 1 of [128], i.e.    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c  2 √ (ln d − ln c) λ∞ ∆2 (c, d)(sup{|ξ  ( cd)|, |ξ  (d)|}) ≤ 8  √ + ∆2 (d, c)(sup{|ξ  (c)|, |ξ  ( cd)|}) .

Integral Inequalities for Strongly Generalized Convex Functions

47

Theorem 2.4.2. Let ξ : K ⊆ R+ = (0, ∞) → R be a differentiable function on K 0 and c, d ∈ K 0 with c < d, and let λ : [c, d] √ → [0, ∞) be1 a continuous positive mapping and geometrically symmetric to cd. If ξ  ∈ √ L [c, d], ξ : K ⊆ R+ = (0, ∞) → R is geometrically symmetric with respect to cd and |ξ  |α is strongly GA-convex on [c, d] for α > 1 with modulus µ > 0, then    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c  1− α1   1− α1 α α (ln d − ln c)2 α−1 ∆2 (c α−1 , d α−1 ) λ∞ ≤ 8 α   α α |ξ  (d)| |ξ (c)| (∆1 (c, d) − ∆5 (c, d)) + (∆1 (c, d) + ∆5 (c, d)) × 2 2 1/α  1− α1 α α µ 2 + ∆2 (d α−1 , c α−1 ) − ln d − ln c (∆1 (c, d) − ∆6 (c, d)) 4  α α |ξ (c)| |ξ  (d)| (∆1 (d, c) + ∆5 (d, c)) + (∆1 (d, c) − ∆5 (d, c)) × 2 2  1/α µ 2 , − ln d − ln c (∆1 (d, c) − ∆6 (d, c)) 4

where λ∞ = supx∈[c,d] |λ(x)|.

Proof From Lemma 2.2.8, we have    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c    1 1 (δ) (ln d − ln c) ln x ≤ λ∞ dx 1 (δ)|ln(1 (δ))||ξ  (1 (δ))|dδ 2(ln d + ln c) x 0 2 (δ)    1  1 (δ) ln x  dx 2 (δ)|ln(2 (δ))||ξ (2 (δ))|dδ . + x 0 2 (δ) Applying H¨ older’s inequality, we have    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c 1− α1   α α α − 1 1 α−1 (ln d − ln c)2 λ∞ 1 (δ)|ln(1 α−1 (δ))|dδ ≤ 8 α 0  1 1/α × |ln(1 (δ))||ξ  (1 (δ))|α dδ 

0

1− α1  α α α − 1 1 α−1 α−1 + 2 (δ)|ln(2 (δ))|dδ α 0  1 1/α  . × |ln(2 (δ))||ξ  (2 (δ))|α dδ 0

48

Integral Inequalities and Generalized Convexity

Using Lemmas 2.2.6 and 2.2.7, and strong GA-convexity of |ξ  |α on [c, d] for α > 1 with modulus µ > 0, we have    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c  1− α1 α α (ln d − ln c)2 α−1 α−1 α−1 λ∞ ∆2 (c ,d ) ≤ 8 α   α α |ξ (c)| |ξ  (d)| (∆1 (c, d) − ∆5 (c, d)) + (∆1 (c, d) + ∆5 (c, d)) × 2 2 1− α1 1/α  α − 1 α α µ 2 α−1 α−1 ∆2 (d + ,c ) − ln d − ln c (∆1 (c, d) − ∆6 (c, d)) 4 α   α α |ξ  (d)| |ξ (c)| (∆1 (d, c) + ∆5 (d, c)) + (∆1 (d, c) − ∆5 (d, c)) × 2 2 1/α  µ 2 − ln d − ln c (∆1 (d, c) − ∆6 (d, c)) . 4 This completes the proof. Corollary 2.4.3. If λ(x) = in Theorem 2.4.2, then

1 (ln x)(ln d−ln c) ,

∀ x ∈ [c, d] with 1 < c < d < ∞

   d  (ln d)ξ(d) + (ln c)ξ(c) 1 ξ(x)   − dx    ln d + ln c ln d − ln c c x

 1− α1  α α 1 (ln d − ln c) α − 1 (∆2 (c α−1 , d α−1 ))1− α ≤ 8 ln c α   α α |ξ  (d)| |ξ (c)| (∆1 (c, d) − ∆5 (c, d)) + (∆1 (c, d) + ∆5 (c, d)) × 2 2 1/α  1− α1 α α µ 2 + ∆2 (d α−1 , c α−1 ) − ln d − ln c (∆1 (c, d) − ∆6 (c, d)) 4   α α |ξ  (d)| |ξ (c)| (∆1 (d, c) + ∆5 (d, c)) + (∆1 (d, c) − ∆5 (d, c)) × 2 2 1/α  µ 2 − ln d − ln c (∆1 (d, c) − ∆6 (d, c)) . 4

Integral Inequalities for Strongly Generalized Convex Functions

49

Corollary 2.4.4. If µ = 0 in Theorem 2.4.2, then    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c 1− α1   1− α1  α α α−1 (ln d − ln c)2 λ∞ ∆2 (c α−1 , d α−1 ) ≤ 8 α   1/α α α |ξ (c)| |ξ  (d)| (∆1 (c, d) − ∆5 (c, d)) + (∆1 (c, d) + ∆5 (c, d)) × 2 2  1− α1 α α + ∆2 (d α−1 , c α−1 )   1/α  α α |ξ  (d)| |ξ (c)| (∆1 (d, c) + ∆5 (d, c)) + (∆1 (d, c) − ∆5 (d, c)) × , 2 2 where λ∞ = supx∈[c,d] |λ(x)|. Remark 2.4.2. If |ξ  |α is geometrically quasi convex, then above theorem reduces to Theorem 2 of [128]: i.e.    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c 1    1− α 1− α1  α α 1 α−1 (ln d − ln c)2 λ∞ (∆1 (c, d)) α ∆2 (c α−1 , d α−1 ) ≤ 8 α √  1− α1 α α 1 × (sup{|ξ  ( cd)|, |ξ  (d)|}) + ∆2 (d α−1 , c α−1 ) (∆1 (d, c)) α  √ × (sup{|ξ  (c)|, |ξ  ( cd)|}) .

Theorem 2.4.3. Let ξ : K ⊆ R+ = (0, ∞) → R be a differentiable function on K 0 and c, d ∈ K 0 with c < d, and let λ : [c, d] √ → [0, ∞) be1 a continuous L [c, d], ξ : K ⊆ positive mapping and geometrically symmetric to cd. If ξ  ∈ √ R+ = (0, ∞) → R is geometrically symmetric with respect to cd and |ξ  |α is strongly GA-convex on [c, d] for α > 1 with modulus µ > 0 and α > l > 0, then    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c 1      1− α 1/α  1− α1 α−l α−l α−1 1 (ln d − ln c)2 λ∞ ∆2 (c α−1 , d α−1 ) ≤ 8 α−l l

50

Integral Inequalities and Generalized Convexity 

α

α

|ξ  (d)| |ξ  (c)| (∆2 (cl , dl ) − ∆3 (cl , dl )) + (∆2 (cl , dl ) + ∆3 (cl , dl )) 2 2 1/α  1− α1 α−l α−l µ 2 + ∆2 (d α−1 , c α−1 ) − ln d − ln c (∆2 (cl , dl ) − ∆4 (cl , dl ) 4  α α |ξ (c)| |ξ  (d)| (∆2 (dl , cl ) + ∆3 (dl , cl )) + (∆2 (dl , cl ) − ∆3 (dl , cl )) × 2 2 1/α  µ 2 l l l l , − ln d − ln c (∆2 (d , c ) − ∆4 (d , c ) 4 ×

where λ∞ = supx∈[c,d] |λ(x)|. Proof From Lemma 2.2.8, we have    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c    1 1 (δ) (ln d − ln c) ln x ≤ λ∞ dx 1 (δ)|ln(1 (δ))||ξ  (1 (δ))|dδ 2(ln d + ln c) x 0 2 (δ)    1  1 (δ) ln x  dx 2 (δ)|ln(2 (δ))||ξ (2 (δ))|dδ . + (2.17) x 0 2 (δ) Applying H¨ older’s inequality in (2.17), we have    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c  1− α1  α−l α−l α − 1 1 α−1 (ln d − ln c)2 α−1 λ∞ 1 (δ)|ln(1 (δ))|dδ ≤ 8 α−l 0   1 1/α 1 × l1 (δ)|ln(l1 (δ))||ξ  (1 (δ))|α dδ l 0  1− α1  α−l α−l α − 1 1 α−1 + 2 (δ)|ln(2 α−1 (δ))|dδ α−l 0   1 1/α  1 l l  α . ×  (δ)|ln(2 (δ))||ξ (2 (δ))| dδ l 0 2 Using Lemmas 2.2.6 and 2.2.7, and strong GA-convexity of |ξ  |α on [c, d] for α > 1 with modulus µ > 0, we have    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c

Integral Inequalities for Strongly Generalized Convex Functions

51

   1/α  α−l α−l 1 α−1 1 (ln d − ln c)2 (∆2 (c α−1 , d α−1 ))1− α λ∞ 8 α−l l   α α |ξ (c)| |ξ  (d)| (∆2 (cl , dl ) − ∆3 (cl , dl )) + (∆2 (cl , dl ) + ∆3 (cl , dl )) × 2 2 1/α  1− α1 α−l α−l µ 2 + ∆2 (d α−1 , c α−1 ) − ln d − ln c (∆2 (cl , dl ) − ∆4 (cl , dl )) 4  α α |ξ (c)| |ξ  (d)| (∆2 (dl , cl ) + ∆3 (dl , cl )) + (∆2 (dl , cl ) − ∆3 (dl , cl )) × 2 2 1/α  µ 2 l l l l − ln d − ln c (∆2 (d , c ) − ∆4 (d , c )) . 4 1 1− α

≤

This completes the proof.

Corollary 2.4.5. If λ(x) = (ln x)(ln1 d−ln c) , ∀ x ∈ [c, d] with 1 < c < d < ∞ in Theorem 2.4.3, then    d  (ln d)ξ(d) + (ln c)ξ(c) 1 ξ(x)   − dx    ln d + ln c ln d − ln c c x  1− α1  1/α  α−l α−l 1 (ln d − ln c) α−1 1 (∆2 (c α−1 , d α−1 ))1− α λ∞ 8(ln c) α−l l   α α |ξ  (d)| |ξ (c)| (∆2 (cl , dl ) − ∆3 (cl , dl )) + (∆2 (cl , dl ) + ∆3 (cl , dl )) × 2 2 1/α  1− α1 α−l α−l µ 2 + ∆2 (d α−1 , c α−1 ) − ln d − ln c (∆2 (cl , dl ) − ∆4 (cl , dl ) 4  α α |ξ (c)| |ξ  (d)| (∆2 (dl , cl ) + ∆3 (dl , cl )) + (∆2 (dl , cl ) − ∆3 (dl , cl )) × 2 2 1/α  µ 2 l l l l . − ln d − ln c (∆2 (d , c ) − ∆4 (d , c ) 4

≤

Corollary 2.4.6. If µ = 0 in Theorem 2.4.3, then    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c 1      1− α 1/α  1− α1 α−l α−l α−1 1 (ln d − ln c)2 λ∞ ∆2 (c α−1 , d α−1 ) ≤ 8 α−l l   1/α α α |ξ (c)| |ξ  (d)| (∆2 (cl , dl ) − ∆3 (cl , dl )) + (∆2 (cl , dl ) + ∆3 (cl , dl )) × 2 2  1− α1 α−l α−l + ∆2 (d α−1 , c α−1 )   1/α  α α |ξ (c)| |ξ  (d)| l l l l l l l l (∆2 (d , c ) + ∆3 (d , c )) + (∆2 (d , c ) − ∆3 (d , c )) × , 2 2 where λ∞ = supx∈[c,d] |λ(x)|.

52

Integral Inequalities and Generalized Convexity

Remark 2.4.3. If |ξ  |α is geometrically quasi convex, then above theorem reduces to Theorem 3 of [128]: i.e.    d  (ln d)ξ(d) + (ln c)ξ(c)  d (ln x)λ(x) (ln x)λ(x)ξ(x)   dx − dx    ln d + ln c x x c c  1− α1  1/α  1− α1 α−l α−l (ln d − ln c)2 α−1 1 ∆2 (c α−1 , d α−1 ) λ∞ ≤ 8 α−l l  1− α1 1 √   α−l α−l × ∆2 (cl , dl ) α (sup{|ξ  ( cd)|, |ξ  (d)|}) + ∆2 (d α−1 , c α−1 )  √  1 × ∆2 (dl , cl ) α (sup{|ξ  (c)|, |ξ  ( cd)|}) .

2.5

Hermite–Hadamard Type Integral Inequalities for the Class of Strongly Convex Functions on Time Scales

In this section, first, we define a class of strongly convex function with respect to two auxiliary functions ψ1 and ψ2 on time scales T [92]. Definition 2.5.1. Consider a time scale T and let ψ1 , ψ2 : (0, 1) → R be two nonnegative funcions. A function ξ : K = [c, d]T → R is said to be a (ψ1 , ψ2 )−strongly convex function with respect to two nonnegative functions ψ1 and ψ2 if there exists a constant µ > 0 such that ξ((1 − δ)c + δd) ≤ ψ1 (1 − δ)ψ2 (δ)ξ(c) + ψ2 (1 − δ)ψ1 (δ)ξ(d)

− µδ(1 − δ)(d − c)2 , ∀ c, d ∈ K, δ ∈ [0, 1].

Now, we discuss some new special cases of Definition 2.5.1. (I). If ψ1 (δ) = ψ2 (δ) = δ s in Definition 2.5.1, then we get Breckner type of s−strongly convex functions. Definition 2.5.2. Consider a time scale T and s ∈ [0, 1] be a real number. A function ξ : K = [c, d]T → R is a Breckner type s−strongly convex function, if ξ((1 − δ)c + δd) ≤ (1 − δ)s δ s [ξ(c) + ξ(d)] − µδ(1 − δ)(d − c)2 , ∀ c, d ∈ K, δ ∈ [0, 1].

(II). If ψ1 (δ) = ψ2 (δ) = 1 in Definition 2.5.1, then we get P −strongly convex functions.

Integral Inequalities for Strongly Generalized Convex Functions

53

Definition 2.5.3. Consider a time scale T, then ξ : K = [c, d]T → R is a P −strongly convex function, if ξ((1 − δ)c + δd) ≤ ξ(c) + ξ(d) − µδ(1 − δ)(d − c)2 , ∀ c, d ∈ K, δ ∈ [0, 1].

Next, we shall present Hermite–Hadamard type inequalities for (ψ1 , ψ2 )– strongly convex functions on Time-scales. [92] Theorem 2.5.1. Consider a time scale T and K = [c, d]T such that c < d and c, d ∈ T. Suppose that there is a delta differentiable function ξ : K → R on K 0 . If |ξ ∆ | is (ψ1 , ψ2 )−strongly convex function with respect to two nonnegative functions ψ1 and ψ2 , then    d   ξ(c) + ξ(d) 1    − ξ (x)∆x    2 d−c c ≤

d−c [(A∗ (δ) + B ∗ (δ))(|ξ ∆ (c)| + |ξ ∆ (d)|) − 2µ(d − c)2 C ∗ (δ)], 2

where ∗

A (δ) = B ∗ (δ) = and



1

δψ1 (δ)ψ2 (1 − δ)∆δ,

0



1

δψ2 (δ)ψ1 (1 − δ)∆δ

0

C ∗ (δ) =



1 0

δ 2 (1 − δ)∆δ.

Proof Using Corollary 2.2.1, modulus property and (ψ1 , ψ2 )−strong convexity of |ξ ∆ |, we obtain    d  ξ(c) + ξ(d)  1   − ξ  (x)∆x    2 d−c c  1   1 d−c ∆ ∆ ≤ δ|ξ (δd + (1 − δ)c)|∆δ + δ|ξ (δc + (1 − δ)d)|∆δ 2 0 0  1 d−c ≤ δ{ψ1 (δ)ψ2 (1 − δ)|ξ ∆ (d)| + ψ2 (δ)ψ1 (1 − δ)|ξ ∆ (c)| 2 0  1 − µδ(1 − δ)(d − c)2 }∆δ + δ{ψ1 (δ)ψ2 (1 − δ)|ξ ∆ (c)| 0

+ ψ2 (δ)ψ1 (1 − δ)|ξ ∆ (d)| − µδ(1 − δ)(d − c)2 }∆δ]  1 d−c δ{ψ1 (δ)ψ2 (1 − δ)(|ξ ∆ (c)| + |ξ ∆ (d)|) + ψ2 (δ)ψ1 (1 − δ)(|ξ ∆ (c)| = 2 0   1 ∆ 2 2 + |ξ (d)|)}∆δ − 2µ(d − c) δ (1 − δ)∆δ 0

d−c [(A∗ (δ) + B ∗ (δ))(|ξ ∆ (c)| + |ξ ∆ (d)|) − 2µ(d − c)2 C ∗ (δ)]. = 2 This completes the proof.

54

Integral Inequalities and Generalized Convexity

Corollary 2.5.1. In Theorem 2.5.1, if |ξ ∆ | is a Breckner type s−strongly convex function, then    d  ξ(c) + ξ(d)  1   − ξ  (x)∆x    2 d−c c  1 ∆ ∆ 2 ≤ (d − c)[H1 (δ)(|ξ (c)| + |ξ (d)|) − 2µ(d − c) δ 2 (1 − δ)∆δ], 0

where

H1 (δ) =



1

δ s+1 (1 − δ)s ∆δ.

0

Corollary 2.5.2. In Theorem 2.5.1, if |ξ ∆ | is a P −strongly convex function, then    d   ξ(c) + ξ(d) 1   − ξ  (x)∆x    2 d−c c  1 ∆ ∆ 2 ≤ (d − c)[H2 (δ)(|ξ (c)| + |ξ (d)|) − 2µ(d − c) δ 2 (1 − δ)∆δ], 0

where

H2 (δ) =



1

δ∆δ. 0

Corollary 2.5.3. If T = R, then delta integral reduces to the usual Riemann integral from calculus. Hence, Theorem 2.5.1 becomes    d   ξ(c) + ξ(d) 1   − ξ(x)dx    2 d−c c     d−c (A∗ (δ) + B ∗ (δ))(|ξ (c)| + |ξ (d)|) − 2µ(d − c)2 C ∗ (δ) , ≤ 2 where  1

A∗ (δ) =

δψ1 (δ)ψ2 (1 − δ)dδ,

0

B ∗ (δ) = and ∗



1

0

C (δ) =

δψ2 (δ)ψ1 (1 − δ)dδ 

1 0

δ 2 (1 − δ)dδ.

Remark 2.5.1. When µ = 0 then above theorem reduces to Theorem 3 of [140], i.e.,    d   ξ(c) + ξ(d) 1    − ξ (x)∆x    2 d−c c ≤

d−c [(A∗ (δ) + B ∗ (δ))(|ξ ∆ (c)| + |ξ ∆ (d)|)]. 2

Integral Inequalities for Strongly Generalized Convex Functions

55

Remark 2.5.2. Letting T = R along with ψ1 (δ) = δ 3 , ψ2 (δ) = 1 and µ = 0 then above theorem reduces to Theorem 2.2 of [47], i.e.,    d  d−c   ξ(c) + ξ(d)   1   − |ξ (c)| + |ξ (d)| . ξ(x)dx ≤    2 d−c c 8 Theorem 2.5.2. Consider a time scale T and K = [c, d]T such that c < d and c, d ∈ T. Suppose that there is a delta differentiable function ξ : K → R on K 0 . If |ξ ∆ |b is (ψ1 , ψ2 )−strongly convex function with respect to two nonnegative functions ψ1 and ψ2 , where a1 + 1b = 1 with b > 1. Then, we have    d   ξ(c) + ξ(d) 1   − ξ  (x)∆x    2 d−c c  1  a1  1 d−c a δ ∆δ {ψ1 (δ)ψ2 (1 − δ)|ξ ∆ (d)|b ≤ 2 0 0 1 + ψ2 (δ)ψ1 (1 − δ)|ξ ∆ (c)|b − µδ(1 − δ)(d − c)2 }∆δ b  1 + {ψ1 (δ)ψ2 (1 − δ)|ξ ∆ (c)|b + ψ2 (δ)ψ1 (1 − δ)|ξ ∆ (d)|b 0 1  −µδ(1 − δ)(d − c)2 }∆δ b .

Proof Using Corollary 2.2.1, modulus property, H¨older’s integral inequality and (ψ1 , ψ2 )−strong convexity of |ξ ∆ |, we get    d   ξ(c) + ξ(d) 1    − ξ (x)∆x    2 d−c c   1   1    d − c  ∆ ∆ +  ≤ δξ (δd + (1 − δ)c)∆δ δξ (δc + (1 − δ)d)∆δ     2 0 0  1 1  1  a  1 b d−c δ a ∆δ |ξ ∆ (δd + (1 − δ)c)|b ∆δ ≤ 2 0 0  1  1b  ∆ b + |ξ (δc + (1 − δ)d)| ∆δ 0

d−c ≤ 2



1 a

δ ∆δ 0

 a1 

1 0

{ψ1 (δ)ψ2 (1 − δ)|ξ ∆ (d)|b

1 + ψ2 (δ)ψ1 (1 − δ)|ξ ∆ (c)|b − µδ(1 − δ)(d − c)2 }∆δ b  1 + {ψ1 (δ)ψ2 (1 − δ)|ξ ∆ (c)|b + ψ2 (δ)ψ1 (1 − δ)|ξ ∆ (d)|b 0 1  −µδ(1 − δ)(d − c)2 }∆δ b .

This completes the proof.

56

Integral Inequalities and Generalized Convexity

Corollary 2.5.4. In Theorem 2.5.2, if |ξ ∆ |b is a Breckner type s−strongly convex function, then    d  ξ(c) + ξ(d)  1    − ξ (x)∆x    2 d−c c  a1  1  1 δ a ∆δ {δ s (1 − δ)s )(|ξ ∆ (c)|b + |ξ ∆ (d)|b ) ≤ (d − c) 0 0 1  − µδ(1 − δ)(d − c)2 }∆δ b . Corollary 2.5.5. In Theorem 2.5.2, if |ξ ∆ |b is a P −strongly convex function, then    d   ξ(c) + ξ(d) 1    − ξ (x)∆x    2 d−c c 1  a  1  1 δ a ∆δ {|ξ ∆ (c)|b + |ξ ∆ (d)|b ≤ (d − c) 0 0 1  − µδ(1 − δ)(d − c)2 }∆δ b .

Corollary 2.5.6. If T = R, then delta integral reduces to the usual Riemann integral from calculus. Hence, Theorem 2.5.2 becomes    d  ξ(c) + ξ(d)  1   − ξ(x)dx    2 d−c c  1  a1  1   d−c a δ dδ {ψ1 (δ)ψ2 (1 − δ)|ξ (d)|b + ψ2 (δ)ψ1 (1 − δ)|ξ (c)|b ≤ 2 0 0  1 1   2 b − µδ(1 − δ)(d − c) }dδ + {(ψ1 (δ)ψ2 (1 − δ))|ξ (c)|b 0  1b   b 2 . + ψ2 (δ)ψ1 (1 − δ)|ξ (d)| − µδ(1 − δ)(d − c) }dδ

Remark 2.5.3. When µ = 0, then above theorem reduces to Theorem 4 of [140], i.e.,    d  ξ(c) + ξ(d)  1    − ξ (x)∆x    2 d−c c  1  a1  1 d−c δ a ∆δ {ψ1 (δ)ψ2 (1 − δ)|ξ ∆ (d)|b ≤ 2 0 0  1 1 +ψ2 (δ)ψ1 (1 − δ)|ξ ∆ (c)|b }∆δ b + {ψ1 (δ)ψ2 (1 − δ)|ξ ∆ (c)|b 0 1  + ψ2 (δ)ψ1 (1 − δ)|ξ ∆ (d)|b }∆δ b .

Integral Inequalities for Strongly Generalized Convex Functions

57

Remark 2.5.4. Letting T = R along with ψ1 (δ) = δ 3 , ψ2 (δ) = 1 and µ = 0 then above theorem reduces to Theorem 2.3 of [47], i.e.,    1b    d    ξ(c) + ξ(d) 1 d−c |ξ (d)|b + |ξ (c)|b   − ξ(x)dx ≤ .   2(a + 1) a1  2 d−c c 2

Theorem 2.5.3. Consider a time scale T and K = [c, d]T such that c < d and c, d ∈ T. Suppose that there is a delta differentiable function ξ : K → R on K 0 . If |ξ ∆ | is (ψ1 , ψ2 )−strongly convex function with respect to two nonnegative functions ψ1 and ψ2 , then      d    c+d 1    − ξ (x)∆x ≤ (d − c) A∗∗ (δ)|ξ ∆ (d)| + B ∗∗ (δ)|ξ ∆ (c)| ξ   2 d−c c  − µ(d − c)2 C ∗∗ (δ) ,

where



∗∗

A (δ) =

B ∗∗ (δ) =

δψ1 (δ)ψ2 (1 − δ)∆δ +



δψ2 (δ)ψ1 (1 − δ)∆δ +



1/2 0



1/2 0

and ∗∗

C (δ) =



1 1/2 1 1/2

1/2 2

0

(1 − δ)ψ1 (δ)ψ2 (1 − δ)∆δ,

δ (1 − δ)∆δ +



(1 − δ)ψ2 (δ)ψ1 (1 − δ)∆δ 1

1/2

δ(1 − δ)2 ∆δ.

Proof Using Corollary 2.2.2, modulus property and (ψ1 , ψ2 )−strong convexity of |ξ ∆ |, we get      d   1 c+d    − ξ (x)∆x ξ   2 d−c c    1 1/2 ≤ (d − c) δ|ξ ∆ (δd + (1 − δ)c)|∆δ + |δ − 1||ξ ∆ (δd + (1 − δ)c)|∆δ 0

≤ (d − c)



0

1/2

1/2

δ{ψ1 (δ)ψ2 (1 − δ)|ξ ∆ (d)| + ψ2 (δ)ψ1 (1 − δ)|ξ ∆ (c)|

− µδ(1 − δ)(d − c)2 }∆δ +



1 1/2

(1 − δ){ψ1 (δ)ψ2 (1 − δ)|ξ ∆ (d)|

+ ψ2 (δ)ψ1 (1 − δ)|ξ ∆ (c)| − µδ(1 − δ)(d − c)2 }∆δ



58

Integral Inequalities and Generalized Convexity

= (d − c)



∆

× |ξ (d)| +

1/2

δψ1 (δ)ψ2 (1 − δ)∆δ +

0





1 1/2

1/2

δψ2 (δ)ψ1 (1 − δ)∆δ +

0

∆

2

×|ξ (c)| − µ(d − c)



(1 − δ)ψ1 (δ)ψ2 (1 − δ)∆δ



1 1/2

1/2 2

0



δ (1 − δ)∆δ +



(1 − δ)ψ2 (δ)ψ1 (1 − δ)∆δ 1 2

1/2

δ(1 − δ) ∆δ

  = (d − c) A∗∗ (δ)|ξ ∆ (d)| + B ∗∗ (δ)|ξ ∆ (c)| − µ(d − c)2 C ∗∗ (δ) .





This completes the proof.

Corollary 2.5.7. If T = R, then our delta integral reduces to the usual Riemann integral from calculus. Hence, Theorem 2.5.3 becomes      d   c+d 1   − ξ(x)dx ξ   2 d−c c

≤ (d − c) [A∗∗ (δ)|ξ  (c)| + B ∗∗ (δ)|ξ  (d)|  − µ(d − c)2 C ∗∗ (δ) ,

where A∗∗ (δ) = B ∗∗ (δ) = and ∗∗

C (δ) =



1/2

δψ1 (δ)ψ2 (1 − δ)dδ +

0





1/2

δψ2 (δ)ψ1 (1 − δ)dδ +

0

1/2 2

0

δ (1 − δ)dδ +





1 1/2



(1 − δ)ψ1 (δ)ψ2 (1 − δ)dδ,

1 1/2

(1 − δ)ψ2 (δ)ψ1 (1 − δ)dδ

1 1/2

δ(1 − δ)2 dδ.

Remark 2.5.5. When µ = 0 then above theorem reduces to Theorem 5 of [140], i.e.,      d     1 c+d   − ξ  (x)∆x ≤ (d − c) A∗∗ (δ)|ξ ∆ (d)| + B ∗∗ (δ)|ξ ∆ (c)| . ξ   2 d−c c

Remark 2.5.6. Letting T = R along with ψ1 (δ) = δ, ψ2 (δ) = 1 and µ = 0 then above theorem reduces to Theorem 2.2 of [86], i.e.,      d  d−c  1 c+d   − [|ξ  (c)| + |ξ  (d)|] . ξ(x)dx ≤ ξ   2 d−c c 8

Integral Inequalities for Strongly Generalized Convex Functions

59

Theorem 2.5.4. Consider a time scale T and K = [c, d]T such that c < d and c, d ∈ T. Suppose that there is a delta differentiable function ξ : K → R on K 0 . If |ξ ∆ |b is (ψ1 , ψ2 )−strongly convex function with respect to two nonnegative functions ψ1 and ψ2 , where a1 + 1b = 1 with b > 1. Then, we have      d   1 c+d    − ξ (x)∆x ξ   2 d−c c  1     a 1/2

≤ (d − c) 

1/2

δ a ∆δ

0

0

{ψ1 (δ)ψ2 (1 − δ)|ξ ∆ (d)|b

+ψ2 (δ)ψ1 (1 − δ)|ξ ∆ (c)|b − µδ(1 − δ)(d − c)2 }∆δ   a1  1

1

+

1/2

|1 − δ|a ∆δ

1/2

 1b

{ψ1 (δ)ψ2 (1 − δ)|ξ ∆ (d)|b

 1 + ψ2 (δ)ψ1 (1 − δ)|ξ (c)| − µδ(1 − δ)(d − c)2 }∆δ ) b . ∆

b

Proof Using Corollary 2.2.2, modulus property, H¨older’s integral inequality and (ψ1 , ψ2 )−strong convexity of |ξ ∆ |, we get      d   1 c+d    − ξ (x)∆x ξ   2 d−c c     1   1/2 ∆    ≤ (d − c)  δξ (δd + (1 − δ)c)∆δ  +  (δ − 1)ξ ∆ (δd + (1 − δ)c)∆δ  

≤ (d − c) +



0

1/2



1/2

a

δ ∆δ

0

1

1/2

|1 − δ|a ∆δ



≤ (d − c) 



 a1 

 a1 

1/2 a

δ ∆δ

0

1/2

∆

|ξ (δd + (1 − δ)c| ∆δ

0

1 1/2

 a1 

1

1/2 0

1/2

1

|1 − δ|a ∆δ

 1b

 1b  |ξ ∆ (δd + (1 − δ)c|b ∆δ  {ψ1 (δ)ψ2 (1 − δ)|ξ ∆ (d)|b

+ψ2 (δ)ψ1 (1 − δ)|ξ ∆ (c)|b − µδ(1 − δ)(d − c)2 }∆δ   a1 

+

b

1/2

 1b

{ψ1 (δ)ψ2 (1 − δ)|ξ ∆ (d)|b

 1 + ψ2 (δ)ψ1 (1 − δ)|ξ ∆ (c)|b − µδ(1 − δ)(d − c)2 }∆δ ) b .

This completes the proof.

60

Integral Inequalities and Generalized Convexity

Corollary 2.5.8. If T = R, then our delta integral reduces to the usual Riemann integral from calculus. Hence, Theorem 2.5.4 becomes      d   1 c+d   − ξ(x)dx ξ   2 d−c c   a1   1/2 1/2 a  δ dδ {ψ1 (δ)ψ2 (1 − δ)|ξ  (d)|b ≤ (d − c) 0

0

+ψ2 (δ)ψ1 (1 − δ)|ξ  (c)|b − µδ(1 − δ)(d − c)2 }dδ   a1  1

+

1/2

1

|1 − δ|a dδ

1/2

 1b

{ψ1 (δ)ψ2 (1 − δ)|ξ  (d)|b

+ ψ2 (δ)ψ1 (1 − δ)|ξ  (c)|b − µδ(1 − δ)(d − c)2 }dδ

 1b 

.

Remark 2.5.7. When µ = 0 then above theorem reduces to Theorem 6 of [140], i.e.,      a1   d  1/2   1 c + d    a  − ξ (x)∆x ≤ (d − c) δ ∆δ ξ   2 d−c c 0 ×

 +

1/2

∆

0



b

∆

b

{ψ1 (δ)ψ2 (1 − δ)|ξ (d)| + ψ2 (δ)ψ1 (1 − δ)|ξ (c)| }∆δ

1 a

1/2

|1 − δ| ∆δ

 a1 

 1b

1 1/2

{ψ1 (δ)ψ2 (1 − δ)|ξ ∆ (d)|b

 1 + ψ2 (δ)ψ1 (1 − δ)|ξ (c)| }∆δ ) b . ∆

b

Remark 2.5.8. Letting T = R along with ψ1 (δ) = δ, ψ2 (δ) = 1 and µ = 0 then above theorem reduces to Theorem 2.3 of [86], i.e.,      d   c+d 1   − ξ(x)dx ξ   2 d−c c   a1   1b   d−c 4 |ξ (d)|b + 3|ξ (c)|b ≤ 16 a+1   1   b b b + |ξ (c)| + 3|ξ (d)| .

Theorem 2.5.5. Let ξ : T → R be a differntiable mapping and c, d ∈ T with c < d. Let |ξ ∆ | be (ψ1 , ψ2 )−strongly convex function with respect to two

Integral Inequalities for Strongly Generalized Convex Functions

61

nonnegative functions ψ1 and ψ2 , then    d   1   ξ  (x)∆x ξ(c){1 − γ2 (1, 0)} + ξ(d)γ2 (1, 0) −   d−c c ≤

d − c ∗∗∗ [A (δ, δ1 )|ξ ∆ (c)| + B ∗∗∗ (δ, δ1 )|ξ ∆ (d)| − µ(d − c)2 C ∗∗∗ (δ, δ1 )], 2

where A∗∗∗ (δ, δ1 ) = B ∗∗∗ (δ, δ1 ) = and C

∗∗∗



1 0



1 0



1

{ψ1 (δ)ψ2 (1 − δ) + ψ1 (δ1 )ψ2 (1 − δ1 )}(δ + δ1 )∆δ∆δ1 ,

0



1 0

(δ, δ1 ) =

{ψ2 (δ)ψ1 (1 − δ) + ψ2 (δ1 )ψ1 (1 − δ1 )}(δ + δ1 )∆δ∆δ1 , 

1 0



1 0

{δ(1 − δ) + δ1 (1 − δ1 )}(δ + δ1 )∆δ∆δ1 .

Proof Using Lemma 2.2.9, property of modulus and (ψ1 , ψ2 )−strong convexity of |ξ ∆ |, we obtain    d   1    ξ (x)∆x ξ(c){1 − γ2 (1, 0)} + ξ(d)γ2 (1, 0) −   d−c c  1 1 d−c ≤ |ξ ∆ (δc + (1 − δ)d) − ξ ∆ (δ1 c + (1 − δ1 )d)||δ − δ1 |∆δ∆δ1 2 0 0   d−c 1 1 ∆ ≤ {|ξ (δc + (1 − δ)d)| + |ξ ∆ (δ1 c + (1 − δ1 )d)|}(δ + δ1 )∆δ∆δ1 2 0 0   d−c 1 1 ≤ {ψ1 (δ)ψ2 (1 − δ)|ξ ∆ (c)| + ψ2 (δ)ψ1 (1 − δ)|ξ ∆ (d)| 2 0 0 − µδ(1 − δ)(d − c)2 + ψ1 (δ1 )ψ2 (1 − δ1 )|ξ ∆ (c)| + ψ2 (δ1 )ψ1 (1 − δ1 )|ξ ∆ (d)| − µδ1 (1 − δ1 )(d − c)2 }(δ + δ1 )∆δ∆δ1 d−c = 2



1 0



1 0

[{ψ1 (δ)ψ2 (1 − δ) + ψ1 (δ1 )ψ2 (1 − δ1 )}|ξ ∆ (c)|

+ {ψ2 (δ)ψ1 (1 − δ) + ψ2 (δ1 )ψ1 (1 − δ1 )}|ξ ∆ (d)|

− µ(d − c)2 {δ(1 − δ) + δ1 (1 − δ1 )}](δ + δ1 )∆δ∆δ1 d − c ∗∗∗ [A (δ, δ1 )|ξ ∆ (c)| + B ∗∗∗ (δ, δ1 )|ξ ∆ (d)| − µ(d − c)2 C ∗∗∗ (δ, δ1 )]. = 2 This completes the proof.

62

Integral Inequalities and Generalized Convexity

Corollary 2.5.9. If T = R, then our delta integral reduces to the usual Riemann integral from calculus. Hence, Theorem 2.5.5 becomes    d   ξ(c) + ξ(d) 1   − ξ(x)dx    2 d−c c ≤

d − c ∗∗∗ [A (δ, δ1 )|ξ  (c)| + B ∗∗∗ (δ, δ1 )|ξ  (d)| − µ(d − c)2 C ∗∗∗ (δ, δ1 )], 2

where A∗∗∗ (δ, δ1 ) = B ∗∗∗ (δ, δ1 ) = C

∗∗∗

and



1 0



1 0



1

{ψ1 (δ)ψ2 (1 − δ) + ψ1 (δ1 )ψ2 (1 − δ1 )}(δ + δ1 )dδdδ1 ,

0



1 0

(δ, δ1 ) =

{ψ2 (δ)ψ1 (1 − δ) + ψ2 (δ1 )ψ1 (1 − δ1 )}(δ + δ1 )dδdδ1 , 

1 0



1 0

{δ(1 − δ) + (δ1 (1 − δ1 )}(δ + δ1 )dδdδ1

γ2 (1, 0) =



1 0

(1 − δ)dδ = 1/2.

Remark 2.5.9. When µ = 0 then above theorem reduces to Theorem 7 of [140], i.e.,    d   1   ξ  (x)∆x ξ(c){1 − γ2 (1, 0)} + ξ(d)γ2 (1, 0) −   d−c c ≤

d − c ∗∗∗ [A (δ, δ1 )|ξ ∆ (c)| + B ∗∗∗ (δ, δ1 )|ξ ∆ (d)|]. 2

Remark 2.5.10. Letting T = R along with ψ1 (δ) = 12 , ψ2 (δ) = 14 and µ = 0 then above theorem reduces to Theorem 2.2 of [47], i.e.,    d  d−c   ξ(c) + ξ(d)   1   − |ξ (c)| + |ξ (d)| . ξ(x)dx ≤    2 d−c c 8 Theorem 2.5.6. Let ξ : [c, d]T → R be a delta differentiable mapping on T0 such that c < d. If |ξ ∆ | is (ψ1 , ψ2 )−strongly convex function with respect to two nonnegative functions ψ1 and ψ2 , then      d   1 c+d    − ξ (x)∆x ξ   2 d−c c ≤

d − c ∗∗∗∗ [A (δ, δ1 )|ξ ∆ (c)| + B ∗∗∗∗ (δ, δ1 )|ξ ∆ (d)| − µ(d − c)2 C ∗∗∗∗ (δ, δ1 )], 2

Integral Inequalities for Strongly Generalized Convex Functions where A∗∗∗∗ (δ, δ1 ) = B ∗∗∗∗ (δ, δ1 ) = and



1 0



1 0



1

|ψ(δ1 )−ψ(δ)| [ψ1 (δ)ψ2 (1 − δ) + ψ1 (δ1 )ψ2 (1 − δ1 )] ∆δ∆δ1 ,

0



C ∗∗∗∗ (δ, δ1 ) =

63

1 0



|ψ(δ1 )−ψ(δ)| [ψ1 (1 − δ)ψ2 (δ) + ψ1 (1 − δ1 )ψ2 (δ1 )] ∆δ∆δ1 1 0



1 0

|ψ(δ1 ) − ψ(δ)| [δ(1 − δ) + δ1 (1 − δ1 )] ∆δ∆δ1 .

Proof Using Lemma 2.2.10, property of modulus and (ψ1 , ψ2 )−strong convexity of |ξ ∆ |, we obtain      d   1 c+d    − ξ (x)∆x ξ   2 d−c c  1 1 d−c ≤ |ξ ∆ (δc + (1 − δ)d) − ξ ∆ (δ1 c + (1 − δ1 )d)| 2 0 0 × |ψ(δ1 ) − ψ(δ)|∆δ∆δ1   d−c 1 1 ∆ ≤ {|ξ (δc + (1 − δ)d)| + |ξ ∆ (δ1 c + (1 − δ1 )d)|} 2 0 0 × |ψ(δ1 ) − ψ(δ)|∆δ∆δ1 ≤

d−c 2



1 0



1 0

{ψ1 (δ)ψ2 (1 − δ)|ξ ∆ (c)| + ψ2 (δ)ψ1 (1 − δ)|ξ ∆ (d)|

− µδ(1 − δ)(d − c)2 + ψ1 (δ1 )ψ2 (1 − δ1 )|ξ ∆ (c)| + ψ2 (δ1 )ψ1 (1 − δ1 )|ξ ∆ (d)|

− µδ1 (1 − δ1 )|d − c|2 }|ψ(δ1 ) − ψ(δ)|∆δ∆δ1   d−c 1 1 = [{ψ1 (δ)ψ2 (1 − δ) + ψ1 (δ1 )ψ2 (1 − δ1 )}|ξ ∆ (c)| 2 0 0 + {ψ2 (δ)ψ1 (1 − δ) + {ψ2 (δ1 )ψ1 (1 − δ1 )}|ξ ∆ (d)|

− µ(d − c)2 {δ(1 − δ) + δ1 (1 − δ1 )}]|ψ(δ1 ) − ψ(δ)|∆δ∆δ1 d − c ∗∗∗∗ [A = (δ, δ1 )|ξ ∆ (c)| + B ∗∗∗∗ (δ, δ1 )|ξ ∆ (d)| − µ(d − c)2 C ∗∗∗∗ (δ, δ1 )]. 2 This completes the proof. Remark 2.5.11. If T = R, then our delta integral reduces to the usual Riemann integral from calculus. Hence, Theorem 2.5.6 becomes      d   c+d 1   − ξ(x)dx ξ   2 d−c c ≤

d − c ∗∗∗∗ [A (δ, δ1 )|ξ  (c)| + B ∗∗∗∗ (δ, δ1 )|ξ  (d)| − µ(d − c)2 C ∗∗∗∗ (δ, δ1 )], 2

64

Integral Inequalities and Generalized Convexity

where A

∗∗∗∗

B

(δ, δ1 ) =

∗∗∗∗

and

(δ, δ1 ) =



1 0



1 0



1

|ψ(δ1 )−ψ(δ)| [ψ1 (δ)ψ2 (1 − δ) + ψ1 (δ1 )ψ2 (1 − δ1 )] dδdδ1 ,

0



1 0

C ∗∗∗∗ (δ, δ1 ) =

|ψ(δ1 )−ψ(δ)| [ψ1 (1 − δ)ψ2 (δ) + ψ1 (1 − δ1 )ψ2 (δ1 )] dδdδ1 

1 0



1 0

|ψ(δ1 ) − ψ(δ)| [δ(1 − δ) + δ1 (1 − δ1 )] dδdδ1 .

Remark 2.5.12. When µ = 0 then above theorem reduces to Theorem 8 of [140], i.e.,      d   1 c+d    − ξ (x)∆x ξ   2 d−c c ≤

d − c ∗∗∗∗ [A (δ, δ1 )|ξ ∆ (c)| + B ∗∗∗∗ (δ, δ1 )|ξ ∆ (d)|]. 2

Remark 2.5.13. Letting T = R along with ψ1 (δ) = 12 , ψ2 (δ) = 14 and µ = 0 then above theorem reduces to Theorem 2.2 of [86], i.e.,      d   d−c 1 c+d   − [|ξ  (c)| + |ξ  (d)|] . ξ(x)dx ≤ ξ   2 d−c c 8

Example 2.5.1. Let T = R. Obviously, ξ(x) = x is a strongly convex function with ψ1 (δ) = 2 − δ, ψ2 (δ) = 1, µ = 1 and continuous on (0, ∞), so we may apply Theorem 2.5.1 with c = 1/2 and d = 1. Clearly    d  ξ(c) + ξ(d)  1   − ξ(x)dx    2 d−c c  1 3 = −2 xdx 4 1/2 = 0.

(2.18)

On the other hand   d−c [(A∗ (δ) + B ∗ (δ))(|ξ (c)| + |ξ (d)|) − 2µ(d − c)2 C ∗ (δ)] 2   1 1 1 9 ×2−2×1× × = 4 6 4 12

≈ 0.7395,

(2.19)

Integral Inequalities for Strongly Generalized Convex Functions where



A∗ (δ) =

C ∗ (δ) =

0



B ∗ (δ) = and

1



δ(2 − δ)dδ =

2 , 3

δ(1 + δ)dδ =

5 6

1 0

1

δ 2 (1 − δ)dδ =

0

65

1 . 12

From (2.18) and (2.19), we see that 0 < 0.7395. Example 2.5.2. Let T = R. Obviously, ξ(x) = x + 1 is strongly convex with ψ1 (δ) = 2, ψ2 (δ) = 4, µ = 1/4 and continuous on (0, ∞), so we may apply Theorem 2.5.3 with c = 0 and d = 1/4. Clearly      d   1 c+d   − ξ(x)dx ξ   2 d−c c    1/2 9    = −4 (x + 1)dx 8  0 = 0.

(2.20)

On the other hand   (d − c) A∗∗ (δ)|ξ  (c)| + B ∗∗ (δ)|ξ  (d)| − c(d − c)2 C ∗∗   1 1 5 1 (2 × 1 + 2 × 1) − × × = 4 4 16 96 ≈ 0.9997,

where A∗∗ (δ) = B ∗∗ (δ) = 8 C ∗∗ (δ) =



1/2 0



1/2

δdδ + 8 0

δ 2 (1 − δ)dδ +



1 1/2

From (2.20) and (2.21), we see that 0 < 0.9997.



1 1/2

(1 − δ)dδ = 2,

δ(1 − δ)2 dδ =

5 . 96

(2.21)

66

Integral Inequalities and Generalized Convexity √ Example 2.5.3. Let T = R. Obviously, ξ(x) = x is strongly convex with ψ1 (δ) = 4, ψ2 (δ) = δ, µ = 1/4, and continuous on (0, ∞), so we may apply Theorem 2.5.5 with c = 2 and d = 4. Clearly      d   c+d 1   − ξ(x)dx ξ   2 d−c c    2+4 1  4√    = − xdx   2 2 2 ≈ 0.0081.

(2.22)

On the other hand d − c ∗∗∗ [A (δ, δ1 )|ξ ∆ (c)| + B ∗∗∗ (δ, δ1 )|ξ ∆ (d)| − µ(d − c)2 C ∗∗∗ (δ, δ1 )] 2   1 1 4 − 2 10 14 1 1 × √ + × − ×4× = 2 3 3 4 4 3 2 2 ≈ 2.0118, (2.23) where ∗∗∗

A

(δ, δ1 ) =

B

∗∗∗



1 0



1 0

(δ, δ1 ) =

and C ∗∗∗ (δ, δ1 ) =



1 0



{4(1 − δ) + 4(1 − δ1 )}(δ + δ1 )dδdδ1 = 

1 0



1

(4δ + 4δ1 )(δ + δ1 )dδdδ1 = 0

10 , 3

14 3

1 0

{δ(1 − δ) + δ1 (1 − δ1 )}(δ + δ1 )dδdδ1 =

From (2.22) and (2.23), we see that 0.0081 < 2.0118.

1 . 3

Chapter 3 Integral Inequalities for Strongly Generalized Convex Functions of Higher Order

3.1

Introduction

Lin and Fukushima [97] introduced strongly convex functions of higher order to simplify the study of mathematical programs with equilibrium constraints. Obviously, strong convexity of higher order is a generalization of strong convexity, the function ξ(x) = x4 is strongly convex of order 4, but not strongly convex of order 2 on R, see [97]. Lin and Fukushima [97] have established that the optimal solution of MPEC under strong convexity of higher order is same as the optimal solution of penalized problem. Further, Lin and Fukushima [97] have shown that the higher order strong convexity of a function is equivalent to higher order strong monotonicity of the gradient map of the function. Antczak [6] introduced the class of exponentially convex functions which can be considered as a significant extension of the convex functions. Exponentially convex functions have applications in various fields such as Mathematical programming, Information geometry, Big data analysis, Machine learning, Statistics, Sequential prediction and Stochastic optimization, see [2, 122, 123, 131]. Awan et al. [11] investigated some other kinds of exponentially convex functions and established several new Hermite–Hadamard type integral inequalities via exponentially convex functions. Noor and Noor [123] defined and introduced some new concepts of the strongly exponentially convex functions with respect to an auxiliary non-negative bifunction and investigated the optimality conditions for the strongly exponentially convex functions. Recently, Rashied et al. [140] established some Trapezoid type inequalities for generalized fractional integral and related inequalities via exponentially convex functions. Further, Rashid et al. [142] derived a new integral identity involving Riemann–Liouville fractional integral and obtained new fractional bounds involving the functions having exponential convexity property. The organization of this chapter is as follows: In Section 3.2, we recall some basic results that are necessary for our main results. In Section 3.3, we DOI: 10.1201/9781003408284-3

67

68

Integral Inequalities and Generalized Convexity

introduce the concept of strongly η-convex functions of higher order, as a generalization of the strongly η-convex functions. We investigate the Hermite– Hadamard and Hermite–Hadamard-Fej´er type inequalities for strongly ηconvex functions of higher order. Some special cases of these results are also investigated in this section. In Section 3.4, we derive Hermite–Hadamard integral inequality for higher order strongly exponentially convex functions. Also, we discuss Reimann-Liouville fractional estimates via strongly exponentially convex functions of higher order. Moreover, some particular cases of the main results are discussed.

3.2

Preliminaries

Lin and Fukushima [97] gave the following concept of strongly convex functions of higher order. Definition 3.2.1. A function ξ : K ⊆ Rn → R is said to be strongly convex with order σ > 0 on a convex set K ⊆ Rn if there exist a constant µ > 0, such that ξ(δx + (1 − δ)y) ≤ δξ(x) + (1 − δ)ξ(y) − µδ(1 − δ)x − yσ , for any x, y ∈ K and any δ ∈ [0, 1]. For σ = 2, the above definition reduces to the strong convexity. The following concept of exponentially convex function is given by Antczak [6]. Definition 3.2.2. A positive function ξ : K −→ R is said to be an exponentially convex function, if eξ(δx+(1−δ)y) ≤ δeξ(x) + (1 − δ)eξ(y) , ∀ x, y ∈ K, δ ∈ [0, 1]. Noor and Noor [123] gave the following concept of strongly exponentially convex function of higher order. Definition 3.2.3. A positive function ξ on the convex set K is said to be higher order strongly exponentially convex functon of order σ > 1 if there exists a constant µ > 0, such that eξ(δx+(1−δ)y) ≤ δeξ(x) +(1−δ)eξ(y) −µδ(1−δ)||y −x||σ , ∀ x, y ∈ K, δ ∈ [0, 1]. Rashid et al. [140] obtained the following corollary.

Integral Inequalities for Strongly Generalized Convex Functions.....

69

Corollary 3.2.1. Let ξ : K = [c, d] → R be an absolutely continuous mapping on (c, d) such that (eξ ) ∈ L1 [c, d]. Then the following equality holds: Γ(α + 1)[Jxα− eξ(c) + Jxα+ eξ(d) ] (x − c)α eξ(c) + (d − x)α eξ(d) − d−c d−c α+1  1 (x − c) = (δ α − 1)eξ(δx+(1−δ)c) ξ  (δx + (1 − δ)c)dδ d−c 0  (d − x)α+1 1 + (1 − δ α )eξ(δx+(1−δ)d) ξ  (δx + (1 − δ)d) dδ. d−c 0 Rashid et al. [142] proposed the following lemma. Lemma 3.2.1. Let α > 0 be a number and let ξ : K = [c, d] −→ R be a differentiable function on (c, d), then  1   d−c 3c + d +(1−δ)c)  α ξ (δ 3c+d 4 Γξ (c, d, α) = + (1 − δ)c dδ δ e ξ δ 16 4 0    1 c+d 3c+d 3c + d c+d + + (1 − δ) dδ (δ α − 1)eξ(δ 2 +(1−δ) 4 ) ξ  δ 2 4 0    1 c+3d c+d c + 3d c+d + + (1 − δ) dδ δ α eξ(δ 4 +(1−δ) 2 ) ξ  δ 4 2 0     1 c + 3d ξ (δd+(1−δ) c+3d α  ) 4 + dδ , (δ − 1)e ξ δd + (1 − δ) 4 0 where  4(α−1) Γ(α + 1) c+3d 1  ξ( 3c+d 4 ) + eξ ( 4 ) − Γξ (c, d, α) = e 2 (d − c)α   3c+d c+d c+3d J(α3c+d )− eξ(c) + J(αc+d )− eξ( 4 ) + J(αc+3d )− eξ( 2 ) + Jdα− eξ( 4 ) . 4

3.3

2

4

Strongly Generalized Convex Functions of Higher Order

Now we are ready to discuss our main results. We define the η-convex and strongly η-convex function of higher order in Rn [106]. Definition 3.3.1. A function ξ : K ⊆ Rn → R is said to be η-convex function with respect to η : ξ(K) × ξ(K) → R, if ξ(δx + (1 − δ)y) ≤ ξ(y) + δη(ξ(x), ξ(y)), ∀x, y ∈ K, δ ∈ [0, 1].

70

Integral Inequalities and Generalized Convexity

Definition 3.3.2. A function ξ : K ⊆ Rn → R is said to be strongly η-convex function of order σ > 0 with respect to η : ξ(K) × ξ(K) → R and modulus µ > 0, if ξ(δx+(1−δ)y) ≤ ξ(y)+δη(ξ(x), ξ(y))−µδ(1−δ)x−yσ , ∀x, y ∈ K, δ ∈ [0, 1]. (3.1) Example 3.3.1. X = R+ , ξ(x) = 4x, η(x, y) = exp(x − y)4 + x. Then, ξ is strongly η-convex function of order 4 with modulus 1. For σ = 2, the above definition reduces to the strongly η-convex function but if σ is not equal to 2, they are different. Example 3.3.2. X = R+ ∪ {0}, ξ(x) = x, η(x, y) = (x − y)4 + x + y. Then, ξ is strongly η-convex function of order 4 with modulus 1 and is not strongly η-convex function of order 2. Remark 3.3.1. If K ⊂ R and x = y in (3.1), then (3.1) reduces to Remark 1.4 of [12]. Theorem 3.3.1. Let ξ : K ⊆ Rn → R be a differentiable strongly η-convex function of order σ > 0. If ξ has minimum at y, then η(ξ(x), ξ(y)) − µx − yσ ≥ 0.

(3.2)

Proof Since ξ has minimum at y, then ∇ξ(y) = 0 and the condition ∇ξ(y), x − y ≥ 0

(3.3)

is satisfied automatically. We know that ξ is strongly η-convex of order σ > 0, then ξ(δx + (1 − δ)y) ≤ ξ(y) + δη(ξ(x), ξ(y)) − µδ(1 − δ)x − yσ . Dividing above inequality by δ and taking limit δ → 0 on both sides, we have ∇ξ(y), x − y ≤ η(ξ(x), ξ(y)) − µx − yσ .

(3.4)

From (3.3) and (3.4), we have η(ξ(x), ξ(y)) − µx − yσ ≥ 0. This completes the proof. Remark 3.3.2. When K ⊂ R and σ = 2, then above theorem reduces to Theorem 1.6 of [12], i.e., η(ξ(x), ξ(y)) − µ(x − y)2 ≥ 0.

Integral Inequalities for Strongly Generalized Convex Functions.....

71

Theorem 3.3.2. Let ξ : [c, d] → R be strongly η-convex function of order σ > 0 with modulus µ > 0. If η(., .) is bounded from above on ξ([c, d])×ξ([c, d]), then    d c+d Mη µ 1 σ ξ − + d − c ≤ ξ(x)dx 2 2 4(σ + 1) d−c c ξ(c) + ξ(d) η(ξ(c), ξ(d)) + η(ξ(d), ξ(c)) µ ≤ + − d − cσ 2 4 6 µ ξ(c) + ξ(d) Mη + − d − cσ , ≤ 2 2 6 where Mη is upper bound of η. Proof Since ξ is strongly η-convex of order σ, then   c+d ξ 2      1 c + d + δ(d − c) 1 c + d − δ(d − c) =ξ + 2 2 2 2   c + d + δ(d − c) ≤ξ 2      1 c + d − δ(d − c) c + d + δ(d − c) µ + η ξ ,ξ − δ σ d − cσ 2 2 2 4   Mη µ σ c + d + δ(d − c) + − δ d − cσ . ≤ξ 2 2 4 This implies     Mη µ σ c + d + δ(d − c) c+d σ − + δ d − c ≤ ξ . ξ 2 2 4 2 Similarly, ξ



c+d 2



−

Mη µ + δ σ d − cσ ≤ ξ 2 4



c + d − δ(d − c) 2



.

By using the change of variable technique, we have  c+d   d  d 2 1 1 ξ(x)dx = ξ(x)dx + ξ(x)dx c+d d−c c d−c c 2    1 1 c + d − δ(d − c) dδ = ξ 2 0 2    1 1 c + d + δ(d − c) + dδ ξ 2 0 2    1  Mη µ σ c+d σ ≥ ξ − + δ d − c dδ 2 2 4 0   Mη µ c+d =ξ − + d − cσ . 2 2 4(σ + 1)

(3.5)

(3.6)

72

Integral Inequalities and Generalized Convexity

We now prove the right hand side of the theorem. Since ξ is strongly η-convex function of order σ > 0, we have ξ(δc + (1 − δ)d) ≤ ξ(d) + δη(ξ(c), ξ(d)) − µδ(1 − δ)d − cσ . Integrating above inequality with respect to δ on [0, 1], we have  1  1 ξ(δc + (1 − δ)d)dδ ≤ [ξ(d) + δη(ξ(c), ξ(d)) − µδ(1 − δ)d − cσ ]dδ, 0

0

1 d−c



1 d−c



d

µ 1 ξ(x)dx ≤ ξ(d) + η(ξ(c), ξ(d)) − d − cσ = P. 2 6

d

µ 1 ξ(x)dx ≤ ξ(c) + η(ξ(d), ξ(c)) − d − cσ = Q. 2 6

c

Similarly,

c

Therefore,  d 1 ξ(x)dx ≤ M in{P, Q} d−c c ξ(c) + ξ(d) η(ξ(c) + ξ(d)) + η(ξ(d), ξ(c)) µ + − d − cσ ≤ 2 4 6 µ ξ(c) + ξ(d) Mη + − d − cσ . ≤ 2 2 6 This completes the proof. Remark 3.3.3. When σ = 2, then above theorem reduces to Theorem 2.1 of [12], i.e.,    d Mη µ 1 c+d − + d − c2 ≤ ξ(x)dx ξ 2 2 12 d−c c ξ(c) + ξ(d) η(ξ(c), ξ(d)) + η(ξ(d), ξ(c)) µ ≤ + − d − c2 2 4 6 µ ξ(c) + ξ(d) Mη + − d − c2 . ≤ 2 2 6 Remark 3.3.4. If we consider µ = 0, then above theorem reduces to Theorem 5 of [54], i.e.,    d Mη 1 c+d − ≤ ξ(x)dx ξ 2 2 d−c c ξ(c) + ξ(d) η(ξ(c), ξ(d)) + η(ξ(d), ξ(c)) + ≤ 2 4 ξ(c) + ξ(d) Mη ≤ + . 2 2

Integral Inequalities for Strongly Generalized Convex Functions.....

73

Now we establish the result on Fej´er type inequality for strongly η-convex function of order σ > 0 [106]. Theorem 3.3.3. Let ξ : [c, d] → R be a strongly η-convex function of order σ, such that η(., .) is bounded above on ξ([c, d]) × ξ([c, d]). Also suppose that ψ : [c, d] → R+ is integrable and symmetric with respect to c+d 2 , then  d  µ d c+d ψ(x)dx + (c + d − 2x)σ ψ(x)dx − Lη (c, d) ξ 2 4 c c   d ξ(c) + ξ(d) d ≤ ξ(x)ψ(x)dx ≤ ψ(x)dx 2 c c  d − µd − c(σ−2) (d − x)(x − c)ψ(x)dx + Rη (c, d), 

c

where d Lη (c, d) = 12 c η(ξ(c + d − x), ξ(x))ψ(x)dx d and Rη (c, d) = η(ξ(c),ξ(d))+η(ξ(d),ξ(c)) (d − x)ψ(x)dx, 2(d−c) c respectively. Proof First, we prove the first pair inequality of the theorem. Since ξ is strongly η-convex function of order σ, then     c+d 1 1 ξ =ξ ((1 − δ)d + δc) + ((1 − δ)c + δd) 2 2 2 1 ≤ ξ((1 − δ)c + δd) + η (ξ((1 − δ)d + δc), ξ((1 − δ)c + δd)) 2 µ − (d − c)(1 − 2δ)σ . 4 Since ψ : [c, d] → R+ is integrable and symmetric with respect to ξ



c+d 2



c

≤ (d − c)

d

ψ(x)dx = (d − c)ψ



0

(d − c) + 2 − =





µ (d − c) 4 d

µ 4



d c

c+d 2



then

1 0

ψ ((1 − δ)c + δd) dδ

ξ ((1 − δ)c + δd) ψ ((1 − δ)c + δd) dδ 1 0



0

η (ξ((1 − δ)d + δc), ξ((1 − δ)c + δd)) ψ((1 − δ)c + δd)dδ 1

(d − c)(1 − 2δ)σ ψ((1 − δ)c + δd)dδ

ξ(x)ψ(x)dx +

c

−

1



c+d 2 ,

1 2



d c

η (ξ(c + d − x), ξ(x)) ψ(x)dx

c + d − 2xσ ψ(x)dx.

74

Integral Inequalities and Generalized Convexity

Next, we prove the second pair inequality of the theorem,  1  d ξ(x)ψ(x)dx = (d − c) ξ(δc + (1 − δ)d)ψ(δc + (1 − δ)d)dδ. c

0

Using the definition of strongly η-convex function of order σ, we have   1  d ξ(x)ψ(x)dx ≤ (d − c) ξ(d) ψ(δc + (1 − δ)d)dδ c

0

+ η(ξ(c), ξ(d)) σ

− µd − c



Similarly,   d  ξ(x)ψ(x)dx ≤ (d − c) ξ(c) c



δψ(δc + (1 − δ)d)dδ

0

1

ψ(δc + (1 − δ)d)dδ





1 0

1

δψ(δc + (1 − δ)d)dδ

0

1

0

ψ(δc + (1 − δ)d)dδ

+ (d − c)[η(ξ(c), ξ(d)) + η(ξ(d), ξ(c))] 

1 0

(3.8)

 δ(1 − δ)ψ(δc + (1 − δ)d)dδ .

From (3.7) and (3.8), we have  d  2 ξ(x)ψ(x)dx ≤ (d − c)[ξ(c) + ξ(d)]

− 2µd − c(σ+1)



1

+ η(ψ(d), ψ(c))

c

(3.7)

δ(1 − δ)ψ(δc + (1 − δ)d)dδ .

0

0

− µd − cσ

1



1 0

δψ(δc + (1 − δ)d)dδ

δ(1 − δ)ψ(δc + (1 − δ)d)dδ.

Applying change of variable technique in above inequality, we have   d [ξ(c) + ξ(d)] d ξ(x)ψ(x)dx ≤ ψ(x)dx 2 c c  [η(ξ(c), ξ(d)) + η(ξ(d), ξ(c))] d + (d − x)ψ(x)dx 2(d − c) c  d (d − x)(x − c)ψ(x)dx. − µd − c(σ−2) c

This completes the proof.

Remark 3.3.5. When µ = 0, then the first pair of inequality of the above theorem reduces to Theorem 7 of [54], i.e.,   d   d 1 d c+d ξ ψ(x)dx − η(ξ(c + d − x), ξ(x))ψ(x)dx ≤ ξ(x)ψ(x)dx. 2 2 c c c

Integral Inequalities for Strongly Generalized Convex Functions.....

75

Remark 3.3.6. When µ = 0, then the second pair of inequality of the above theorem reduces to Theorem 6 of [54], i.e., 

d c

ξ(x)ψ(x)dx ≤

ξ(c) + ξ(d) 2



d

ψ(x)dx c

η(ξ(c), ξ(d)) + η(ξ(d), ξ(c)) + 2(d − c)



c

d

(d − x)ψ(x)dx.

Now, we discuss a new variant of Hermite–Hadamard inequality for differentiable strongly η-convex of order σ [106]. Theorem 3.3.4. Let ξ : K 0 ⊂ R → R be n-times differentiable strongly ηconvex function of order σ on K 0 where c, d ∈ K 0 with c < d and ξ  ∈ L1 [c, d]. If |ξ (n) |p is strongly η-convex function of order σ with µ ≥ 1, then for n ≥ 2 and p ≥ 1, we have    d n−1  ξ(c) + ξ(d)   (k − 1)(d − c)k 1   − ξ (k) (c) ξ(x)dx −    2 d−c c 2(k + 1)! k=2

(d − c)n (1− p1 ) ≤ α (n)[α(n)|ξ (n) (d)|p + β(n)η(|ξ (n) (c)|p , |ξ (n) (d)|p ) 2n! 1 − µγ(n)c − dσ ] p ,

where α(n) :=

n−1 n+1 ,

β(n) :=

n2 −2 (n+1)(n+2) ,

γ(n) :=

n−1 (n+1)(n+3) .

Proof Recall Lemma 2.2.5  d n−1  (k − 1)(d − c)k 1 ξ(c) + ξ(d) − ξ (k) (c) ξ(x)dx − 2 d−c c 2(k + 1)! k=2  (d − c)n 1 n−1 δ (n − 2δ)ξ (n) (δc + (1 − δ)d)dδ. = 2n! 0 Case 1 When p = 1, using the definition of strong η-convexity of order σ, we have    d n−1   ξ(c) + ξ(d)  (k − 1)(d − c)k 1   (k) − ξ (c) ξ(x)dx −    2 d−c c 2(k + 1)! k=2   1 (d − c)n δ n−1 (n − 2δ)|ξ (n) (δc + (1 − δ)d)|dδ ≤ 2n! 0

76

Integral Inequalities and Generalized Convexity   1 (d − c)n (n) ≤ |ξ (d)| δ n−1 (n − 2δ)dδ 2n! 0  1 (n) (n) + η(|ξ (c)|, |ξ (d)|) δ n (n − 2δ)dδ − µc − dσ =



0

1

δ n (1 − δ)(n − 2δ)dδ

0



n − 1 (n) n2 − 2 |ξ (d)| + η(|ξ (n) (c)|, |ξ (n) (d)|) n+1 (n + 1)(n + 2)  µ(n − 1) σ c − d . − (n + 1)(n + 3)

(d − c) 2n!

n



Case 2 When p > 1,    d n−1   ξ(c) + ξ(d)  (k − 1)(d − c)k 1   (k) − ξ (c) ξ(x)dx −    2 d−c c 2(k + 1)! k=2  1  1 1 (d − c)n (δ n−1 (n − 2δ))1− p (δ n−1 (n − 2δ)) p |ξ (n) (δc + (1 − δ)d)|dδ . ≤ 2n! 0 Using H¨ older’s inequality, we have    d n−1  ξ(c) + ξ(d)   (k − 1)(d − c)k 1   (k) − ξ (c) ξ(x)dx −    2 d−c c 2(k + 1)! k=2

≤



1− p1

1 (d − c)n δ n−1 (n − 2δ)dδ 2n! 0  1  p1 n−1 (n) p × δ (n − 2δ)|ξ (δc + (1 − δ)d)| dδ 0

(d − c)n ≤ 2n!



n−1 n+1

1− p1 

|ξ (n) (d)|p

+ η(|ξ (n) (c)|p , |ξ (n) (d)|p ) − µc − d

σ





1 0



1 0

δ n (n − 2δ)dδ

1 n

0

δ n−1 (n − 2δ)dδ

δ (1 − δ)(n − 2δ)dδ

 p1

.

Integral Inequalities for Strongly Generalized Convex Functions.....

77

This implies    d n−1   ξ(c) + ξ(d)  (k − 1)(d − c)k 1   − ξ (k) (c) ξ(x)dx −    2 d−c c 2(k + 1)! k=2  1− p1  (d − c)n n − 1 n − 1 (n) |ξ (d)|p ≤ 2n! n+1 n+1 n2 − 2 η(|ξ (n) (c)|p , |ξ (n) (d)|p ) + (n + 1)(n + 2)  p1 µ(n − 1) c − dσ . − (n + 1)(n + 3) This completes the proof.

3.4

Integral Inequalities for Higher Order Strongly Exponentially Convex Functions

We derive new Hermite–Hadamard inequality for higher order strongly exponentially convex functions. Theorem 3.4.1. Let ξ : K = [c, d] −→ R be a strongly exponentially convex function of order σ > 1 with modulus µ > 0, then the function satisfies the following: ξ( c+d 2 )

e

µ 1 + d − cσ ≤ 4 d−c



d c

eξ(x) dx ≤

µ eξ(c) + eξ(d) − d − cσ . (3.9) 2 6

Proof Since ξ is a strongly exponentially convex function of order σ > 1 on K, we have eξ(δx+(1−δ)y) ≤ δeξ(x) +(1−δ)eξ(y) −µδ(1−δ)||y −x||σ , ∀ x, y ∈ K, δ ∈ [0, 1]. (3.10) For δ = 12 , we get eξ(

x+y 2 )

≤

µ eξ(x) + eξ(y) − y − xσ . 2 4

Let x = (1 − δ)c + δd and y = δc + (1 − δ)d, we have eξ(

c+d 2 )

≤

µ eξ[(1−δ)c+δd] + eξ[(δc+(1−δ)d)] − d − cσ . 2 4

78

Integral Inequalities and Generalized Convexity

Integrating above with respect to δ over [0, 1] and using the change of variable technique, we have  d c+d 1 µ eξ( 2 ) ≤ eξ(x) dx − d − cσ . (3.11) d−c c 4 Integrating (3.10) with respect to δ over [0, 1], we have  d µ eξ(c) + eξ(d) 1 − d − cσ . eξ(x) dx ≤ d−c c 2 6

(3.12)

From (3.11) and (3.12), we obtain  d µ µ 1 eξ(c) + eξ(d) ξ( c+d ) σ 2 − d − cσ . + d − c ≤ eξ(x) dx ≤ e 4 d−c c 2 6 This completes the proof.

1 Example 3.4.1. Let K = [ 12 , 1] and µ = 10 . Let ξ : K −→ R be defined by ξ(x) = x for all x ∈ K. Obviously, ξ is higher order strongly exponentially 1 . Then, the function ξ satisfies the above theorem. convex function for µ = 10

Remark 3.4.1. When µ = 0, then Theorem 3.4.1 reduces to the following:  d c+d 1 eξ(c) + eξ(d) eξ( 2 ) ≤ , eξ(x) dx ≤ d−c c 2

which is Hermite–Hadamard inequality for exponentially convex functions given by [46]. Now, we obtain some new Riemann–Liouville fractional estimates via strongly exponentially convex functions of higher order. Theorem 3.4.2. Let ξ : K = [c, d] → R be an absolutely continuous mapping on (c, d) such that (eξ ) ∈ L1 [c, d]. If the function |ξ| is strongly exponentially convex function of order σ1 > 1 with modulus µ1 > 0 and |ξ  | is strongly convex function of order σ2 > 0 with modulus µ2 > 0, then    (x − c)α eξ(c) + (d − x)α eξ(d) Γ(α + 1)[Jxα− eξ(c) + Jxα+ eξ(d) ]   −     d−c d−c   (x − c)α+1 + (d − x)α+1 α φ(x) ≤ 3(α + 3) d−c   3 2 α + 6α + 11α (x − c)α+1 φ(c) + (d − x)α+1 φ(d) + 3(α + 1)(α + 2)(α + 3) d−c   2 α+1 α + 5α ∆1 (x, c) + (d − x)α+1 ∆1 (x, d) (x − c) + 6(α + 2)(α + 3) d−c 2 α + 7α − (µ1 |ξ  (x)|d − cσ1 + µ2 |eξ(x) |d − cσ2 ) 12(α + 3)(α + 4)

Integral Inequalities for Strongly Generalized Convex Functions..... 79   α3 + 9α2 + 26α (x − c)α+1 + (d − x)α+1 − × d−c 12(α + 2)(α + 3)(α + 4)     α+1 |ξ (c)|(x − c) + |ξ  (d)|(d − x)α+1 × µ1 d − cσ1 d−c   ξ(c) α+1 + |eξ(d) |(d − x)α+1 |e |(x − c) σ2 +µ2 d − c d−c   3 2 α + 12α + 47α (x − c)α+1 + (d − x)α+1 σ1 +σ2 + µ1 µ2 d − c , 30(α + 3)(α + 4)(α + 5) d−c where ∆1 (x, d)=|eξ(x) ξ  (d)| + |eξ(d) ξ  (x)| ∆1 (x, c)= |eξ(x) ξ  (c)| + |eξ(c) ξ  (x)|, ξ(x)  ξ(c)  ξ (x)|, φ(c)=|e ξ (c)|, φ(d)=|eξ(d) ξ  (d)|. and φ(x)=|e Proof Using Corollary 3.2.1, property of modulus and the given hypothesis of the theorem, we obtain    (x − c)α eξ(c) + (d − x)α eξ(d) Γ(α + 1)[Jxα− eξ(c) + Jxα+ eξ(d) ]   −     d−c d−c  (x − c)α+1 1 α ≤ |(δ − 1)||eξ(δx+(1−δ)c) ξ  (δx + (1 − δ)c)|dδ d−c 0  (d − x)α+1 1 + |(1 − δ α )||eξ(δx+(1−δ)d) ξ  (δx + (1 − δ)d) |dδ d−c 0  (x − c)α+1 1 ≤ (1 − δ α )(δ|eξ(x) | + (1 − δ)|eξ(c) | − µ1 δ(1 − δ)d − cσ1 ) d−c 0 (d − x)α+1 × (δ|ξ  (x)| + (1 − δ)|ξ  (c)| − µ2 δ(1 − δ)d − cσ2 )dδ + d−c  1 × (1 − δ α )(δ|eξ(x) | + (1 − δ)|eξ(d) | − µ1 δ(1 − δ)d − cσ1 )(δ|ξ  (x)| 0

+ (1 − δ)|ξ  (d)| − µ2 δ(1 − δ)d − cσ2 )dδ   1  1 (x − c)α+1 ξ(x)  α 2 ξ(c)  |e ξ (x)| (1 − δ )δ dδ + |e ξ (c)| (1 − δ α )(1 − δ)2 dδ = d−c 0 0  1 +(|eξ(x) ξ  (c)| + |eξ(c) ξ  (x)|) (1 − δ α )δ(1 − δ)dδ − (µ1 |ξ  (x)|d − cσ1 +µ2 |eξ(x) |d − cσ2 ) +µ2 |eξ(c) |d − cσ2 )



0

1

0  1

(1 − δ α )δ 2 (1 − δ)dδ − (µ1 |ξ  (c)|d − cσ1

(1 − δ α )δ(1 − δ)2 dδ + µ1 µ2 d − cσ1 +σ2 0   1 α 2 × (1 − δ )δ (1 − δ)2 dδ 0

80

Integral Inequalities and Generalized Convexity   1 (d − x)α+1 ξ(x)  + |e ξ (x)| (1 − δ α )δ 2 dδ d−c 0  1 +|eξ(d) ξ  (d)| (1 − δ α )(1 − δ)2 dδ + (|eξ(x) ξ  (d)| + |eξ(d) ξ  (x)|) × ×





0

1

0 1

0

(1 − δ α )δ(1 − δ)dδ − (µ1 |ξ  (x)|d − cσ1 + µ2 |eξ(x) |d − cσ2 ) (1 − δ α )δ 2 (1 − δ)dδ − (µ1 |ξ  (d)|d − cσ1 + µ2 |eξ(d) |d − cσ2 )

  1 (1 − δ α )δ(1 − δ)2 dδ + µ1 µ2 d − cσ1 +σ2 (1 − δ α )δ 2 (1 − δ)2 dδ 0 0   α (x − c)α+1 + (d − x)α+1 = φ(x) 3(α + 3) d−c   3 2 α + 6α + 11α (x − c)α+1 φ(c) + (d − x)α+1 φ(d) + 3(α + 1)(α + 2)(α + 3) d−c   2 α+1 α + 5α ∆1 (x, c) + (d − x)α+1 ∆1 (x, d) (x − c) + 6(α + 2)(α + 3) d−c 2 α + 7α (µ1 |ξ  (x)|d − cσ1 + µ2 |eξ(x) |d − cσ2 ) − 12(α + 3)(α + 4)   α3 + 9α2 + 26α (x − c)α+1 + (d − x)α+1 × − d−c 12(α + 2)(α + 3)(α + 4)     α+1 + |ξ  (d)|(d − x)α+1 |ξ (c)|(x − c) σ1 × µ1 d − c d−c   ξ(c) α+1 |e |(x − c) + |eξ(d) |(d − x)α+1 σ2 +µ2 d − c d−c   α3 + 12α2 + 47α (x − c)α+1 + (d − x)α+1 + µ1 µ2 d − cσ1 +σ2 . 30(α + 3)(α + 4)(α + 5) d−c ×



1

This completes the proof. Corollary 3.4.1. If we choose α = 1, then under the assumption of Theorem 3.4.2, we have a new result    d  (x − c)eξ(c) + (d − x)eξ(d)  1   ξ(x) − e dx    d−c d−c c     (x − c)2 + (d − x)2 1 (x − c)2 φ(c) + (d − x)2 φ(d) 1 φ(x) + ≤ 12 d−c 4 d−c   2 2 1 1 (x − c) ∆1 (x, c) + (d − x) ∆2 (x, d) − (µ1 |ξ  (x)|d − cσ1 + 12 d−c 30

Integral Inequalities for Strongly Generalized Convex Functions.....   1 (x − c)2 + (d − x)2 − {µ1 d − cσ1 + µ2 |eξ(x) |d − cσ2 ) d−c 20    |ξ (c)|(x − c)α+1 + |ξ  (d)|(d − x)α+1 × d−c   ξ(c) |e |(x − c)α+1 + |eξ(d) |(d − x)α+1 σ2 +µ2 d − c d−c   1 (x − c)2 + (d − x)2 σ1 +σ2 + µ1 µ2 d − c . 60 d−c

81

Remark 3.4.2. When µ1 , µ2 = 0, then above theorem reduces to Corollary 2.5 of [146], i.e.,    (x − c)α eξ(c) + (d − x)α eξ(d) Γ(α + 1)[Jxα− eξ(c) + Jxα+ eξ(d) ]   −     d−c d−c   (x − c)α+1 + (d − x)α+1 α3 + 6α2 + 11α α φ(x) + ≤ 3(α + 3) d−c 3(α + 1)(α + 2)(α + 3)  α2 + 5α (x − c)α+1 φ(c) + (d − x)α+1 φ(d) + d−c 6(α + 2)(α + 3)   α+1 α+1 ∆1 (x, c) + (d − x) ∆1 (x, d) (x − c) . × d−c ×



Remark 3.4.3. When µ1 , µ2 = 0 and α = 1 then above theorem reduces to Corollary 2.4 of [146], i.e.,    d   (x − c)eξ(c) + (d − x)eξ(d) 1   ξ(x) − e dx    d−c d−c c     2 2 (x − c) + (d − x) 1 (x − c)2 φ(c) + (d − x)2 φ(d) 1 1 φ(x) + + ≤ 12 d−c 4 d−c 12   (x − c)2 ∆1 (x, c) + (d − x)2 ∆1 (x, d) . × d−c Theorem 3.4.3. Let ξ : K → R be an absolutely continuous mapping on (c, d) such that (eξ ) ∈ L1 [c, d], where c, d ∈ K with c < d. If the function |ξ|q is strongly exponentially convex function of order σ1 > 1 with modulus µ1 > 0 and |ξ  |q is strongly convex function of order σ2 > 0 with modulus µ2 > 0,

82

Integral Inequalities and Generalized Convexity

where

1 p

+

1 q

= 1 with q > 1. Then, we have

   (x − c)α eξ(c) + (d − x)α eξ(d) Γ(α + 1)[Jxα− eξ(c) + Jxα+ eξ(d) ]   −     d−c d−c   p1    1 (x − c)α+1 ∆2 (x, c) ∆3 (x, c) 1 β p + 1, + ≤ α α d−c 3 6

µ1 d − cσ1 (|ξ  (x)|q + |ξ  (c)|q ) µ2 d − cσ2 (|eξ(x) |q + |eξ(c) |q ) − 12 12  q1  σ1 +σ2 α+1 µ1 µ2 d − c ∆2 (x, d) ∆3 (x, d) (d − x) + + + 30 d−c 3 6

−

µ1 d − cσ1 (|ξ  (x)|q + |ξ  (d)|q ) µ2 d − cσ2 (|eξ(x) |q + |eξ(d) |q ) − 12 12 1  σ1 +σ2 q µ1 µ2 d − c + , 30 −

where ∆2 (x, c)= |eξ(x) ξ  (x)|q + |eξ(c) ξ  (c)|q , ∆2 (x, d)=|eξ(x) ξ  (x)|q + |eξ(d) ξ  (d)|q ∆3 (x, c)=|eξ(x) ξ  (c)|q + |eξ(c) ξ  (x)|q and ∆3 (x, d)=|eξ(x) ξ  (d)|q + |eξ(d) ξ  (x)|q . Proof Using Corollary 3.2.1, H¨ older’s inequality and the given hypothesis of the theorem, we obtain    α ξ(c) Γ(α + 1)[Jxα− eξ(c) + Jxα+ eξ(d) ]  + (d − x)α eξ(d)  (x − c) e −     d−c d−c 1  p  1  q1  1 (x − c)α+1 α p ξ(δx+(1−δ)c)  q ≤ |(δ − 1)| dδ |e ξ (δx + (1 − δ)c)| dδ d−c 0 0  p1  1  q1  1 (d − x)α+1 + |(1 − δ α )|p dδ |eξ(δx+(1−δ)d) ξ  (δx + (1 − δ)d) |q dδ d−c 0 0  p1  1  1 α+1 (x − c) ≤ |(1 − δ α )|p dδ (1 − δ α )(δ|eξ(x) |q + (1 − δ)|eξ(c) |q d−c 0 0 1 −µ1 δ(1 − δ)d − cσ1 )(δ|ξ  (x)|q + (1 − δ)|ξ  (c)|q − µ2 δ(1 − δ)d − cσ2 )dδ q  p1  1  1 (d − x)α+1 + |(1 − δ α )|p (δ|eξ(x) |q + (1 − δ)|eξ(d) |q d−c 0 0 1 −µ1 δ(1 − δ)d − cσ1 )(δ|ξ  (x)|q + (1 − δ)|ξ  (d)|q − µ2 δ(1 − δ)d − cσ2 )dδ q  p1   1  1 (x − c)α+1 = (1 − δ α )p dδ δ 2 dδ + |eξ(c) ξ  (c)|q |eξ(x) ξ  (x)|q d−c 0 0  1  1 × (1 − δ)2 dδ + (|eξ(x) ξ  (c)|q + |eξ(c) ξ  (x)|q ) δ(1 − δ)dδ 0

0

Integral Inequalities for Strongly Generalized Convex Functions..... − (µ1 |ξ  (x)|q d − cσ1 + µ2 |eξ(x) |q d − cσ2 ) − (µ1 |ξ  (c)|q d − cσ1 + µ2 |eξ(c) |q d − cσ2 )



1 0 1

0

83

δ 2 (1 − δ)dδ

δ(1 − δ)2 dδ

 p1  1 (d − x)α+1 σ1 +σ2 2 2 α p + µ1 µ2 d − c δ (1 − δ) dδ + (1 − δ ) dδ d−c 0 0   1  1 × |eξ(x) ξ  (x)|q δ 2 dδ + |eξ(d) ξ  (d)|q (1 − δ)2 dδ + (|eξ(x) ξ  (d)|q 

1



1

q

0

+|eξ(d) ξ  (x)|q ) × × × =





1 0 1 0



1 0



0

1

0

δ(1 − δ)dδ − (µ1 |ξ  (x)|q d − cσ1 + µ2 |eξ(x) |q d − cσ2 )

δ 2 (1 − δ)dδ − (µ1 |ξ  (d)|q d − cσ1 + µ2 |eξ(d) |q d − cσ2 ) δ(1 − δ)2 dδ + µ1 µ2 d − cσ1 +σ2 (1 − δ α )p dδ

 p1 

α+1

(x − c) d−c





1 0

δ 2 (1 − δ)2 dδ

 q1

∆2 (x, c) ∆3 (x, c) + 3 6

(µ1 |ξ  (x)|q d − cσ1 + µ2 |eξ(x) |q d − cσ2 ) − 12 (µ1 |ξ  (c)|q d − cσ1 + µ2 |eξ(c) |q d − cσ2 ) µ1 µ2 d − cσ1 +σ2 + 12 30  (d − x)α+1 ∆2 (x, d) ∆3 (x, d) + + d−c 3 6

−

−

 q1

(µ1 |ξ  (x)|q d − cσ1 + µ2 |eξ(x) |q d − cσ2 ) 12

(µ1 |ξ  (d)|q d − cσ1 + µ2 |eξ(d) |q d − cσ2 ) µ1 µ2 d − cσ1 +σ2 − + 12 30 =



 q1

  1   p (x − c)α+1 ∆2 (x, c) ∆3 (x, c) 1 1 β p + 1, + α α d−c 3 6

µ1 d − cσ1 (|ξ  (x)|q + |ξ  (c)|q ) µ2 d − cσ2 (|eξ(x) |q + |eξ(c) |q ) − 12 12  q1  σ1 +σ2 α+1 (d − x) µ1 µ2 d − c ∆2 (x, d) ∆3 (x, d) + + + 30 d−c 3 6

−

µ2 d − cσ2 (|eξ(x) |q + |eξ(d) |q ) µ1 d − cσ1 (|ξ  (x)|q + |ξ  (d)|q ) − 12 12  q1  σ1 +σ2 µ1 µ2 d − c + . 30 −

 

84

Integral Inequalities and Generalized Convexity

Corollary 3.4.2. If we choose α = 1, then under the assumption of Theorem 3.4.3, we have a new result    d  (x − c)eξ(c) + (d − x)eξ(d)  1   ξ(x) − e dx    d−c d−c c  p1    (x − c)2 ∆2 (x, c) ∆3 (x, c) 1 + ≤ p+1 d−c 3 6 µ1 d − cσ1 (|ξ  (x)|q + |ξ  (c)|q ) µ2 d − cσ2 (|eξ(x) |q + |eξ(c) |q ) − 12 12  q1  σ1 +σ2 2 µ1 µ2 d − c ∆2 (x, d) ∆3 (x, d) (d − x) + + + 30 d−c 3 6

−

µ1 d − cσ1 (|ξ  (x)|q + |ξ  (d)|q ) µ2 d − cσ2 (|eξ(x) |q + |eξ(d) |q ) − 12 12 1  σ1 +σ2 q µ1 µ2 d − c + . 30 −

Remark 3.4.4. When µ1 , µ2 = 0, then above theorem reduces to Corollary 2.8 of [146], i.e.,    (x − c)α eξ(c) + (d − x)α eξ(d) Γ(α + 1)[Jxα− eξ(c) + Jxα+ eξ(d) ]   −     d−c d−c 1   p1     1 (x − c)α+1 ∆2 (x, c) ∆3 (x, c) q 1 β p + 1, + ≤ α α d−c 3 6   1 (d − x)α+1 ∆2 (x, d) ∆3 (x, d) q + + . d−c 3 6 Remark 3.4.5. When µ1 , µ2 = 0 and α = 1, then above theorem reduces to Corollary 2.7 of [146], i.e.,    d   (x − c)eξ(c) + (d − x)eξ(d) 1   ξ(x) − e dx    d−c d−c c 1  p1     (x − c)2 ∆2 (x, c) ∆3 (x, c) q 1 + ≤ p+1 d−c 3 6 1   (d − x)2 ∆2 (x, d) ∆3 (x, d) q + + . d−c 3 6 Theorem 3.4.4. Let α > 0 be a number and let ξ : K = [c, d] −→ R be a differentiable function on (c, d). If the function |ξ| is strongly exponentially

Integral Inequalities for Strongly Generalized Convex Functions.....

85

convex function of order σ1 > 1 with modulus µ1 > 0 and |ξ  | is strongly convex function of order σ2 > 0 with modulus µ2 > 0. Then, we have |Γξ (c, d, α)|  3     ξ( 3c+d )  3c + d  d−c α + 9α2 + 20α + 6 e 4 ξ  ≤  16 3(α + 1)(α + 2)(α + 3)  4     c+3d c + 3d  + eξ( 4 ) ξ   4    d−c 2  ξ(c)   + e ξ (c) 16 (α + 1)(α + 2)(α + 3)      3 α α + 3α2 + 2α + 6  ξ( c+d c + d   ξ(d)   2 )ξ e + ξ (d) + e   3(α + 3) 3(α + 1)(α + 2)(α + 3)  2 α3 + 9α2 + 38α + 24 (A1 (c, d) + A6 (c, d)) − (A2 (c, d) + (α + 2)(α + 3) 12(α + 2)(α + 3)(α + 4) α3 + 9α2 + 14α + 24 2A3 (c, d) +A7 (c, d)) − − A5 (c, d) (α + 2)(α + 3)(α + 4) 12(α + 2)(α + 3)(α + 4) α(α + 5) α(α + 7) + (A4 (c, d) + A8 (c, d)) − A9 (c, d) 6(α + 2)(α + 3) 12(α + 3)(α + 4)     (d − c) σ1 +σ2 4(α3 + 12α2 + 47α + 30)   µ1 µ2  , + 30(α + 3)(α + 4)(α + 5) 4 

where

    ξ(c)  3c + d   ξ( 3c+d )    + e 4 ξ (c) ,  A1 (c, d) = e ξ  4  σ1        (d − c)    3c + d   (d − c) σ2 ξ( 3c+d )       4 A2 (c, d) = µ1  ξ |,  + µ2  4  |e 4   4 σ1  σ2    (d − c)   (d − c)   ξ(c)    A3 (c, d) = µ1   4  |ξ (c)| + µ2  4  |e |,        ξ( 3c+d )  c + d   ξ( c+d )  3c + d   + e 2 ξ ,  A4 (c, d) = e 4 ξ    2 4  σ1    σ2      (d − c)    c + d      + µ2  (d − c)  |eξ( c+d 2 ) |, A5 (c, d) = µ1    4  ξ  2 4         ξ( c+d )  c + 3d   ξ( c+3d )  c + d   + e 4 ξ ,  A6 (c, d) = e 2 ξ    4 2  σ1   σ2      (d − c)    c + 3d      + µ2  (d − c)  |eξ( c+3d 4 ) |, A7 (c, d) = µ1    4  ξ  4 4      c+3d c + 3d  A8 (c, d) = |eξ( 4 ) ξ  (d)| + eξ(d) ξ  , 4   σ2     (d − c) σ1    |ξ (d)| + µ2  (d − c)  |eξ(d) |. A9 (c, d) = µ1   4   4 

86

Integral Inequalities and Generalized Convexity

Proof Using Lemma 3.2.1, property of modulus and the given hypothesis of the theorem, 4

|Γξ (c, d, α)| ≤ where  I1 =

d−c Ii , 16 i=1

     3c+d 3c + d + (1 − δ)c  dδ δ α eξ(δ 4 +(1−δ)c) ξ  δ 4 0     1     d − c  σ1  ξ( 3c+d α ) ξ(c)   4 ≤ δ e δ  + (1 − δ)|e | − µ1 δ(1 − δ)  4  0          3c + d   d − c  σ2      × δ ξ dδ  + (1 − δ)|ξ (c)| − µ2 δ(1 − δ)  4  4      1  3c+d 3c + d  1 α+2 ξ(c)  = eξ( 4 ) ξ  δ dδ + |e ξ (c)| δ α (1 − δ)2 dδ  4 0 0     1  3c + d  ξ( 3c+d 4 ) ξ  (c)| + |e + eξ(c) ξ  δ α+1 (1 − δ)dδ  4    σ2  0       (d − c) σ1   3c + d    ξ  + µ2  (d − c)  |eξ( 3c+d 4 )| − µ1    4    4  4  σ1       1  (d − c)   (d − c) σ2 ξ(c) α+2      × δ (1 − δ)dδ − µ1  |ξ (c)| + µ2  |e | 4  4  0    1   (d − c) σ1 +σ2 1 α+2  × δ α+1 (1 − δ)2 dδ + µ1 µ2  δ (1 − δ)2 dδ  4  0 0      1  ξ( 3c+d 2 3c + d   ξ(c)   4 )ξ = e + ξ (c) e   (α + 1)(α + 2)(α + 3) α+3 4 1

A2 (c, d) 2A3 (c, d) A1 (c, d) − − (α + 2)(α + 3) (α + 3)(α + 4) (α + 2)(α + 3)(α + 4)    (d − c) σ1 +σ2 2  + µ1 µ2  ,  4  (α + 3)(α + 4)(α + 5)

+

    ξ(δ c+d +(1−δ) 3c+d )  3c + d  c+d  2 4 I2 = + (1 − δ) (1 − δ ) e ξ δ  dδ 2 4 0     ξ( c+d )  c + d  α(α2 + 6α + 11) α e 2 ξ + ≤  3(α + 1)(α + 2)(α + 3) 3(α + 3)  2     3c+d α(α + 5) 3c + d  × eξ( 4 ) ξ  + A4 (c, d)  4 6(α + 2)(α + 3) α(α2 + 9α + 26) α(α + 7) A5 (c, d) − A2 (c, d) − 12(α + 3)(α + 4) 12(α + 2)(α + 3)(α + 4)    (d − c) σ1 +σ2 2α(α2 + 12α + 47)  µ1 µ2  + ,  4  30(α + 3)(α + 4)(α + 5) 

1

α

Integral Inequalities for Strongly Generalized Convex Functions..... 87    1   c+3d c+d c + d  c + 3d I3 = + (1 − δ) δ α eξ(δ 4 +(1−δ) 2 ) ξ  δ  dδ 4 2 0        ( c+d )  c + d  1  ξ( c+3d 2 c + 3d  e 2 ξ  4 )ξ ≤ e +  (α + 1)(α + 2)(α + 3)   α+3 4 2 A6 (c, d) A7 (c, d) 2A5 (c, d) + − − (α + 2)(α + 3) (α + 3)(α + 4) (α + 2)(α + 3)(α + 4)    (d − c) σ1 +σ2 2   + µ1 µ2  (α + 3)(α + 4)(α + 5) 4 

and

    c+3d c + 3d  (1 − δ α ) eξ(δd+(1−δ) 4 ) ξ  δd + (1 − δ)  dδ 4 0       ξ( c+3d )  c + 3d  α(α2 + 6α + 11) α  ξ(d)   e 4 ξ  ≤ e ξ (d) +  3(α + 3) 3(α + 1)(α + 2)(α + 3)  4 α(α + 5) α(α + 7) + A8 (c, d) − A9 (c, d) 6(α + 2)(α + 3) 12(α + 3)(α + 4) α(α2 + 9α + 26) − A7 (c, d) 12(α + 2)(α + 3)(α + 4)    (d − c) σ1 +σ2 2α(α2 + 12α + 47)   + µ1 µ2  . 30(α + 3)(α + 4)(α + 5) 4 

I4 =



1

This completes the proof.

Corollary 3.4.3. If we choose α = 1, then under the assumption of Theorem 3.4.4, we have a new result    d   1  3c+d  1   ξ( 4 ) ξ( c+3d ) ξ(x) +e 4 e dx] −  e  2 d−c c        d − c 1  ξ( 3c+d 3c + d   ξ( c+3d c + 3d    ) ) 4 4 ≤ e ξ ξ  + e  16 2  4 4        1  ξ(c)   1  ξ( c+d 1  ξ(d)   c + d  + + e ξ (c) + e 2 ) ξ  e ξ (d)  12 6 2 12 1 1 A3 (c, d) + (A1 (c, d) + A6 (c, d)) − (A2 (c, d) + A7 (c, d)) − 12 10 30 1 1 1 − A5 (c, d) + (A4 (c, d) + A8 (c, d)) − A9 (c, d) 15 12 30 σ1 +σ2     1 (d − c)  . + µ1 µ2  .  4  10

88

Integral Inequalities and Generalized Convexity

Theorem 3.4.5. Let α > 0 be a number and let ξ : K = [c, d] −→ R be a differentiable function on (c, d). If the function |ξ|q is strongly exponentially convex function of order σ1 > 1 with modulus µ1 > 0 and |ξ  |q is strongly convex function of order σ2 > 0 with modulus µ2 > 0 , where p1 + 1q = 1, q > 1, then |Γξ (c, d, α)|

  p1    p1     ξ( 3c+d )  3c + d q 1 α   4 20 e ≤ ξ 1  1 + pα 4 (60 q )16 α d−c

1       (d − c) σ1 +σ2 q  ξ(c)  q  + e ξ (c) + 10B1 (c, d) − 5B2 (c, d) + 2µ1 µ2   4           ξ( c+3d )  c + 3d q  ξ( c+d )  c + d q   + 10B3 (c, d)  e 2 ξ + 20 e 4 ξ +    4 2  1       p1  (d − c) σ1 +σ2 q  1   + β p + 1, −5B4 (c, d) + 2µ1 µ2   4  α q         c+d  ξ( 3c+d )  3c + d q c + d   + 10B5 (c, d) e 4 ξ × 20 eξ( 2 ) ξ  +    2 4 1     (d − c) σ1 +σ2 q   ξ(d)  q   + 20 e ξ (d) −5B6 (c, d) + 2µ1 µ2  4       ξ( c+3d )  c + 3d q  + 10B7 (c, d) − 5B8 (c, d)  4 + e ξ  4  1     (d − c) σ1 +σ2 q   , +2µ1 µ2   4  

where

  q  q  3c + d   ξ( 3c+d  4 ) ξ  (c) , B1 (c, d) = eξ(c) ξ  + e  4        (d − c) σ1   3c + d q   + |ξ  (c)|q ξ B2 (c, d) = µ1    4   4 σ2      (d − c)  q  ξ( 3c+d  + µ2  e 4 )  + |eξ(c) |q ,  4     q    c+d  ξ( c+3d )  c + d q c + 3d   , e 4 ξ B3 (c, d) = eξ( 2 ) ξ  +    4 2

Integral Inequalities for Strongly Generalized Convex Functions.....           (d − c) σ1   c + 3d q   c + d q   + ξ  ξ B4 (c, d) = µ1     4    4 2 σ2   q  c+d q   (d − c)     ξ( c+3d   + µ2  e 4 )  + eξ( 2 )  ,  4         ξ( 3c+d )  c + d q  ξ( c+d )  3c + d q   ,   4 2 B5 (c, d) = e ξ ξ  + e  2 4 σ1    q      (d − c)    3c + d q   c+d        B6 (c, d) = µ1   + ξ  ξ 4  2 4 σ2        (d − c)  q  3c+d q  ξ( c+d  + µ2  e 2 )  + eξ( 4 )  ,  4    q  c+3d   ξ( 4 )  q  ξ(d)  c + 3d  B7 (c, d) = e ξ (d) + e ξ  , 4 σ1        (d − c)    c + 3d q q    ξ B8 (c, d) = µ1  (d)| + |ξ   4   4  σ2     (d − c)  c+3d  + µ2  |eξ(d) |q + |eξ( 4 ) |q .  4 

89

Proof Using Lemma 3.2.1, H¨ older’s inequality and the given hypothesis of the theorem, we obtain |Γξ (c, d, α)|      p1   p1  2 4 1 1 1 1 d−c q q α p α p (δ ) dδ Jr + (1 − δ ) dδ Jr , ≤ 16 0 0 r=1 r=3 where

q   1   ξ(δ 3c+d +(1−δ)c)  3c + d  dδ e 4 + (1 − δ)c δ J1 = ξ   4 0  q  1    q 1  3c+d 3c + d    ≤ eξ( 4 ) ξ  δ 2 dδ + eξ(c) ξ  (c) (1 − δ)2 dδ  4 0 0    1   ξ(c)  3c + d q ξ( 3c+d )  q   4 + e ξ ξ (c)| δ(1 − δ)dδ  + |e 4 σ1    0  σ2   q   (d − c)    3c + d   (d − c)  ξ( 3c+d ) q       4 − µ1  ξ |  + µ2  4  |e 4   4   σ2      1   (d − c) σ1    |ξ (c)|q + µ2  (d − c)  |eξ(c) |q δ 2 (1 − δ)dδ − µ1  ×  4   4  0 σ1 +σ2  1   1  (d − c)   × δ(1 − δ)2 dδ + µ1 µ2  δ 2 (1 − δ)2 dδ  4  0

0

90

Integral Inequalities and Generalized Convexity  q  q   1  3c+d 1  ξ(c)  q 1  ξ(c)  3c + d  3c + d  = eξ( 4 ) ξ  e + ξ (c) + ξ e    3 4 3 6  4       σ q   (d − c)  1   3c + d  3c+d 1    µ1  +|eξ( 4 ) ξ  (c)|q −   4  ξ 12 4 σ2  σ1      (d − c)   (d − c)  1 ξ( 3c+d  q   4 ) |q − µ1  +µ2   4  |e  4  |ξ (c)| 12       (d − c) σ2 ξ(c) q  (d − c) σ1 +σ2 1     +µ2  |e | + µ1 µ2  4  30 4    q      3c+d 1 3c + d   ξ(c)  q 20 eξ( 4 ) ξ  = + ξ (c) e  + 10B1 (c, d)  60 4     (d − c) σ1 +σ2   , −5B2 (c, d) + 2µ1 µ2  4 

 q   ξ(δ c+3d +(1−δ) c+d )  c + d  c + 3d e 4 2 J2 = + (1 − δ) δ ξ  dδ  4 2 0   q       c+3d  ξ( c+d )  c + d q 1 c + 3d   + 10B3 (c, d) e 2 ξ ≤ 20 eξ( 4 ) ξ  +    60 4 2 σ1 +σ2    (d − c)   −5B4 (c, d) + 2µ1 µ2  ,  4  

1

 q   ξ(δ c+d +(1−δ) 3c+d )  3c + d  c+d e 2 4 + (1 − δ) δ ξ  dδ  2 4 0     q     c+d  ξ( 3c+d )  3c + d q 1 c + d   + 10B5 (c, d) e 4 ξ ≤ 20 eξ( 2 ) ξ  +    60 2 4 σ1 +σ2    (d − c)   −5B6 (c, d) + 2µ1 µ2   4 

J3 =



1

and

  q  ξ(δd+(1−δ) c+3d )  c + 3d  e 4 δd + (1 − δ) ξ  dδ  4 0   q  c+3d  c + 3d q  1    + 10B7 (c, d) ≤ 20 eξ(d) ξ  (d) + eξ( 4 ) ξ   60 4 σ1 +σ2    (d − c)   . −5B8 (c, d) + 2µ1 µ2   4 

J4 =



1

This completes the proof.

Integral Inequalities for Strongly Generalized Convex Functions.....

91

Corollary 3.4.4. If we choose α = 1, then under the assumption of Theorem 3.4.5, we have a new result    d  1  3c+d   1   ξ( 4 ) ξ( c+3d ) ξ(x) 4 − +e e dx]  e 2  d−c c   p1       ξ( 3c+d )  3c + d q  ξ(c)  q 1 d−c   4 20 e ≤ ξ 1  + e ξ (c) 4 (60 q )16 1 + p 1    (d − c) σ1 +σ2 q  +10B1 (c, d) − 5B2 (c, d) + 2µ1 µ2   4  q         c+3d  ξ( c+d )  c + d q c + 3d   + 10B3 (c, d) e 2 ξ + 20 eξ( 4 ) ξ  +    4 2  1        (d − c) σ1 +σ2 q  ξ( c+d )  c + d q    e 2 ξ + 20 −5B4 (c, d) + 2µ1 µ2    4  2  q    3c+d 3c + d  + eξ( 4 ) ξ   + 10B5 (c, d) 4 1    (d − c) σ1 +σ2 q  −5B6 (c, d) + 2µ1 µ2   4    q  c+3d  c + 3d q     + 10B7 (c, d) + 20 eξ(d) ξ  (d) + eξ( 4 ) ξ   4  1    (d − c) σ1 +σ2 q   . −5B8 (c, d) + 2µ1 µ2  4 

Remark 3.4.6. When µ1 , µ2 = 0, then above theorem reduces to Theorem 2.2 of [142], i.e., |Γξ (c, d, α)|

  p1     p1      ξ( 3c+d )  3c + d q 1 α   4 20 e ≤ ξ 1  1 + pα 4 (60 q )16 α  q   q1   c+3d  c + 3d q    + eξ(c) ξ  (c) + 10B1 (c, d) + 20 eξ( 4 ) ξ   4  q   p1  q1      c+d c + d  1 + eξ( 2 ) ξ  (c, d) + 10B + β p + 1, 3  2 α   q  q   q1    ξ( c+d )  c + d   ξ( 3c+d )  3c + d   + e 4 ξ  + 10B5 (c, d) × 20 e 2 ξ    2 4   q     q1    c + 3d   ξ(d)  q  ξ( c+3d )  + 20 e ξ (d) + e 4 ξ .  + 10B7 (c, d) 4 d−c

92

Integral Inequalities and Generalized Convexity

Remark 3.4.7. When µ1 , µ2 = 0 and α = 1, then above theorem reduces to Corollary 2.2 of [142], i.e.,    d  1  3c+d   1   ξ( 4 ) ξ( c+3d ) ξ(x) +e 4 e dx] −  e 2  d−c c   p1       ξ( 3c+d )  3c + d q  ξ(c)  q 1 d−c   4 20 e ≤ ξ 1  + e ξ (c) 4 (60 q )16 1 + p q         ξ( c+d )  c + d q  c+3d 1 c + 3d   e 2 ξ + +10B1 (c, d)) q + 20 eξ( 4 ) ξ     4 2           ξ( c+d )  c + d q  ξ( 3c+d )  3c + d q 1 q     2 4 +10B3 (c, d)) + 20 e ξ ξ  + e  2 4       q q  c+3d 1 c + 3d    +10B5 (c, d)) q + 20 eξ(d) ξ  (d) + eξ( 4 ) ξ   4  1 +10B7 (c, d)) q .

Chapter 4 Integral Inequalities for Generalized Preinvex Functions

4.1

Introduction

Pini [135] established the relationship between invexity and generalized convexity and showed that ξ(x) = x3 is quasi-convex but not invex, since x = 0 is a stationary point but not a minimum point and also given an example ξ(x, y) = −x2 + xy − ey which is invex but not quasi-convex because it fails to satisfy the second-order necessary and sufficient condition for quasi-convexity. Noor proved some Hermite–Hadamard type inequalities for preinvex [119], log-preinvex [118] and the product of two preinvex functions [120]. Further, I¸scan [64] obtained some Hermite–Hadamard type inequalities using fractional integrals for preinvex functions. Recently, Wang and Feˇckan [171] investigated fractional integral identities for a differentiable mapping involving Riemann–Liouville fractional integrals and Hadamard fractional integrals and gave some inequalities via standard convex, r-convex, s-convex, m-convex, (s,m)-convex, (β, m)-convex functions, etc. Further, the Hermite–Hadamard type inequality for fractional integrals obtained by Liu et al. [98]. Bector et al. [15] introduced preunivex and univex functions with respect to a vector function η, scalar function φ and b as generalizations of preinvex, invex, b-preinvex, b-invex and b-vex functions. It is well known that if ξ : K → R is differentiable and preunivex with respect to η, φ and b then ξ is univex with respect to η, φ and b, where b(x, u) = limλ→0+ b(x, u, δ) [15]. Antczak [7] showed the relationships between different classes of (p, r)invex functions with respect to the same function η. It is well known that if a real-valued function ξ defined on an invex set K ⊂ Rn with respect to η is preunivex with respect to η, φ and b then the level set Lα of ξ is invex with respect to η for each α [7]. Dragomir et al. [42] generalized the Hermite–Hadamard type integral inequalities and discussed their application in special means and numerical integration. Recently, Zhang et al. [182] established some new bounds on Hermite– Hadamard and Simpson’s inequalities for mappings whose absolute values of second derivatives are generalized (m, h)-preinvex. They also derived a general

DOI: 10.1201/9781003408284-4

93

94

Integral Inequalities and Generalized Convexity

k-fractional integral identity along with multi parameters for twice differentiable mappings. The organization of this chapter is as follows: In Section 4.2, we recall some basic results which are necessary for our main results. In Section 4.3, we prove a new form of Hermite–Hadamard inequality using left and right-sided ψ-Riemann–Liouville fractional integrals for preinvexity. We present two essential results of ψ-Riemann–Liouville fractional integral identities using the first-order derivative of a preinvex function. With the help of these results, we obtain some fractional Hermite–Hadamard inequalities and give some examples which satisfy the theorems. Further, in Section 4.4, we discuss some applications for special means with the help of these results. In Section 4.5, we propose the concept of generalized (m, h)-preunivex and establish some new bounds on Hermite–Hadamard and Simpson’s inequalities for mappings whose absolute values of second derivatives are generalized (m, h)-preunivex. Some interesting results are also obtained by using the special parameter values for various suitable choices of function h.

4.2

Preliminaries

In this section, we recall some basic definitions and results required for this chapter. Let K ⊆ Rn be non empty, η : K × K → Rn , b : K × K × [0, 1] → R+ , ξ : K → R and φ : R → R. For x ∈ K, y ∈ K, 0 ≤ δ ≤ 1, we assume that b stands for b(x, y, δ) ≥ 0, and δb ≤ 1. Noor [118] established the following Hermite–Hadamard inequalities for preinvex functions. Theorem 4.2.1. Let ξ : K = [c, c + η(d, c)] → (0, ∞) be a preinvex function on the interval of real numbers int(K) and c, d ∈ int(K) with c < c + η(d, c). Then,    c+η(d,c) 1 ξ(c) + ξ(d) 2c + η(d, c) ≤ . (4.1) ξ(x)dx ≤ ξ 2 η(d, c) c 2 The following definition is taken by Kilbas et al. [85]. Definition 4.2.1. Let (c, d) (−∞ ≤ c < d ≤ ∞) be an interval of the real line R and α > 0. Also let ψ(x) be an increasing and positive monotone function on (c, d], having a continuous derivative ψ  (x) on (c, d). The left and rightsided ψ-Riemann–Liouville fractional integrals of a function ξ with respect to an other function ψ on [c, d] are defined by  x 1 α:ψ ψ  (v)(ψ(x) − ψ(v))α−1 ξ(v)dv, Jc+ ξ(x) = Γ(α) c

Integral Inequalities for Generalized Preinvex Functions  d 1 α:ψ Jd− ξ(x) = ψ  (v)(ψ(v) − ψ(x))α−1 ξ(v)dv, Γ(α) x respectively.

95

The following concept of preunivex functions established by Bector et al. [15]. Definition 4.2.2. At y ∈ K, where K is a invex set, the function ξ is said to be preunivex with respect to η, φ and b, if for every x ∈ K and 0 ≤ δ ≤ 1, ξ[y + δη(x, y)] ≤ ξ(y) + δbφ[ξ(x) − ξ(y)]. Du et al. [48] introduced the following concept of m-invex set. Definition 4.2.3. A set K ⊆ Rn is called m-invex with respect to the mapping η : K × K × (0, 1] → Rn for some fixed m ∈ (0, 1] if mx + δη(y, x, m) ∈ K holds for all x, y ∈ K and δ ∈ [0, 1]. Zhang et al. [182] derived the following general k-fractional integral identity with multi-parameters for twice differentiable functions. Lemma 4.2.1. Let K ⊆ R be an open m-invex subset with respect to η : K × K × (0, 1] → R\{0} for some fixed m ∈ (0, 1], and let c, d ∈ A, c < d with η(d, c, m) > 0. Assume that ξ : K → R is a twice differentiable function  on K such that ξ is integrable on [mc, mc + η(d, c, m)]. Then the following identity for Riemann–Liouville k-fractional integrals along with x ∈ [c, d], λ ∈ [0, 1], α > 0, and k > 0 exists: Iξ,η (α, k; x, λ, m, c, d)  1  α    α η k +2 (x, c, m) α + 1 λ − δ k ξ (mc + δη(x, c, m))dδ δ = α ( k + 1)η(d, c, m) 0 k  1  α α    +2 +2 α (−1) k η k (x, d, m) α + 1 λ − δ k ξ (md + δη(x, d, m))dδ, δ + α ( k + 1)η(d, c, m) k 0

where

α 1−λ [η k (x, c, m)ξ(mc + η(x, c, m)) η(d, c, m) α α α λ [η k (x, c, m)ξ(mc) + (−1) k η k (x, d, m)ξ(md + η(x, d, m))] + η(d, c, m)

Iξ,η (α, k; x, λ, m, c, d) =

α

α

+ (−1) k η k (x, d, m)ξ(md)] 1

+

α k +1

−λ

η(d, c, m) α

α

α



[(−1) k +1 η k +1 (x, d, m)ξ (md + η(x, d, m)) 

− η k +1 (x, c, m)ξ (mc + η(x, c, m))] − +

α k J(md+η(x,d,m))+ ξ(md)]

and Γk is the k-gamma function.

Γk (α + k) α [k J(mc+η(x,c,m)) − ξ(mc) η(d, c, m)

96

4.3

Integral Inequalities and Generalized Convexity

Hermite–Hadamard Type Inequalities via Preinvex Functions

In this section, first, we prove Hermite–Hadamard inequalities for ψRiemann–Liouville fractional integrals via preinvexity [158]. Theorem 4.3.1. Let K ⊆ R be an open invex subset with respect to η : K × K → R and c, d ∈ K with c < c + η(d, c). If ξ : [c, c + η(d, c)] → (0, ∞) is a preinvex function, ξ ∈ L1 [c, c + η(d, c)] and η satisfies Condition C. Also suppose ψ(x) is an increasing and positive monotone function on (c, c + η(d, c)), having a continuous derivative ψ  (x) on (c, c + η(d, c)) and α ∈ (0, 1). Then,   1 Γ(α + 1)  α:ψ ξ c + η(d, c) ≤ α J −1 + (ξ o ψ)ψ −1 (c + η(d, c)) 2 2η (d, c) ψ (c)  −1 + Jψα:ψ (c) −1 (c+η(d,c))− (ξ o ψ)ψ ≤

ξ(c) + ξ(d) ξ(c) + ξ(c + η(d, c)) ≤ . 2 2

Proof By definition of invex set x, y ∈ K, then x + δη(y, x) ∈ K, ∀δ ∈ [0, 1]. From Theorem 4.2.1, we get   1 ξ(x) + ξ(y) ξ x + η(y, x) ≤ . (4.2) 2 2 Using x = c + (1 − δ)η(d, c) and y = c + δη(d, c) in (4.2), we have 1 ξ(c + (1 − δ)η(d, c) + η(c + δη(d, c), c + (1 − δ)η(d, c))) 2 ξ(c + (1 − δ)η(d, c)) + ξ(c + δη(d, c)) ≤ . 2 Applying Condition C in (4.3), we have   1 ξ(c + (1 − δ)η(d, c)) + ξ(c + δη(d, c)) ξ c + η(d, c) ≤ . 2 2

(4.3)

(4.4)

Multiplying (4.4) by δ α−1 on both sides and integrating the resultant with respect to δ over [0, 1], we have    1 2 1 ξ c + η(d, c) ≤ δ α−1 ξ(c + (1 − δ)η(d, c))dδ α 2 0  1 + δ α−1 ξ(c + δη(d, c))dδ. (4.5) 0

Integral Inequalities for Generalized Preinvex Functions

97

Next,  Γ(α + 1)  α:ψ −1 Jψ−1 (c)+ (ξ o ψ)ψ −1 (c + η(d, c)) + Jψα:ψ (c) −1 (c+η(d,c))− (ξ o ψ)ψ α 2η (d, c)  −1 ψ (c+η(d,c)) α (c + η(d, c) − ψ(v))α−1 (ξ o ψ)(v)ψ  (v)dv = α 2η (d, c) ψ−1 (c)   ψ−1 (c+η(d,c)) α−1  + (ψ(v) − c) (ξ o ψ)(v)ψ (v)dv ψ −1 (c)

=

α 2



1

δ α−1 ξ(c + (1 − δ)η(d, c))dδ +

0



0

1

 δ α−1 ξ(c + δη(d, c))dδ .

(4.6)

From (4.5) and (4.6), we have   1 Γ(α + 1)  α:ψ −1 ξ c + η(d, c) ≤ α J (c + η(d, c)) + (ξ o ψ)ψ 2 2η (d, c) ψ−1 (c)  −1 + Jψα:ψ (c) . −1 (c+η(d,c))− (ξ o ψ)ψ

Now, we prove the second pair inequality of the theorem

ξ(c + δη(d, c)) = ξ(c + η(d, c) + (1 − δ)η(c, c + η(d, c))) ≤ δξ(c + η(d, c)) + (1 − δ)ξ(c).

(4.7)

ξ(c + (1 − δ)η(d, c)) = ξ(c + η(d, c) + δη(c, c + η(d, c))) ≤ (1 − δ)ξ(c + η(d, c)) + δξ(c).

(4.8)

Similarly,

From (4.7) and (4.8), we have ξ(c + δη(d, c)) + ξ(c + (1 − δ)η(d, c)) ≤ ξ(c) + ξ(c + η(d, c)).

(4.9)

Multiplying both sides by δ α−1 in (4.9), then integrating with respect to δ over 0 to 1, we have  1  1 δ α−1 ξ(c + δη(d, c))dδ + δ α−1 ξ(c + (1 − δ)η(d, c))dδ 0

0

ξ(c) + ξ(c + η(d, c)) . ≤ α

(4.10)

From (4.6) and (4.10), we have  Γ(α + 1)  α:ψ −1 Jψ−1 (c)+ (ξ o ψ)ψ −1 (c + η(d, c)) + Jψα:ψ (c) −1 (c+η(d,c))− (ξ o ψ)ψ α 2η (d, c) ξ(c) + ξ(d) ξ(c) + ξ(c + η(d, c)) ≤ ≤ . 2 2 This completes the proof.

98

Integral Inequalities and Generalized Convexity

Remark 4.3.1. When η(d, c) = d−c, then above theorem reduces to Theorem 2.1 of [98], i.e.   Γ(α + 1)  α:ψ c+d ξ J −1 + (ξ o ψ)(ψ −1 (d)) ≤ 2 2(d − c)α ψ (c)  ξ(c) + ξ(d) −1 + Jψα:ψ . (c)) ≤ −1 (d)− (ξ o ψ)(ψ 2

Now, we present results of ψ-Riemann–Liouville fractional integral identities including the first-order derivative of a preinvex function [158]. Lemma 4.3.1. Let K ⊆ R be an open invex subset with respect to η : K×K → R and c, d ∈ K with c < c + η(d, c). Suppose that ξ : K → R is a differentiable function. If ξ  is preinvex function on K and ξ  ∈ L1 [c, c + η(d, c)], ψ(x) is an increasing and positive monotone function on (c, c + η(d, c)), having a continuous derivative ψ  (x) on (c, c + η(d, c)) and α ∈ (0, 1). Then, ξ(c) + ξ(c + η(d, c)) Γ(α + 1)  α:ψ − α J −1 + (ξ o ψ)ψ −1 (c + η(d, c)) 2 2η (d, c) ψ (c)  −1 (ξ o ψ)ψ (c) + Jψα:ψ −1 (c+η(d,c))−  ψ−1 (c+η(d,c)) 1 = α [(ψ(v) − c)α − (c + η(d, c) − ψ(v))α ] 2η (d, c) ψ−1 (c) × (ξ  o ψ)(v)ψ  (v)dv  η(d, c) 1 ((1 − δ)α − δ α )ξ  (c + (1 − δ)η(d, c))dδ. = 2 0

Γ(α+1) α:ψ −1 Proof Let M1 = 2η (c + η(d, c)) α (d,c) Jψ −1 (c)+ (ξ o ψ)ψ and Γ(α+1) α:ψ −1 M2 = 2η (c). α (d,c) Jψ −1 (c+η(d,c))− (ξ o ψ)ψ

α M1 = α 2η (d, c)



1 =− α 2η (d, c)

ψ −1 (c+η(d,c))

ψ −1 (c)



(c + η(d, c) − ψ(v))α−1 (ξ o ψ)(v)ψ  (v)dv

ψ −1 (c+η(d,c))

ψ −1 (c)

(ξ o ψ)(v)d(c + η(d, c) − ψ(v))α

  ψ−1 (c+η(d,c)) 1 −ξ(c)η α (d, c) − =− α (c + η(d, c) − ψ(v))α 2η (d, c) ψ −1 (c)

× (ξ  o ψ)(v)ψ  (v)dv]  ψ−1 (c+η(d,c)) 1 ξ(c) + α (c + η(d, c) − ψ(v))α (ξ  o ψ)(v)ψ  (v)dv. = 2 2η (d, c) ψ−1 (c)

Integral Inequalities for Generalized Preinvex Functions M2 =



α α 2η (d, c)

ψ

−1

(c+η(d,c))

ψ −1 (c)

99

(ψ(v) − c)α−1 (ξ o ψ)(v)ψ  (v)dv

 ψ−1 (c+η(d,c)) 1 (ξ o ψ)(v)d(ψ(v) − c)α 2η α (d, c) ψ−1 (c) 1 [ξ(c + η(d, c))η α (d, c) = α 2η (d, c)   ψ−1 (c+η(d,c)) (ψ(v) − c)α (ξ  o ψ)(v)ψ  (v)dv − =

ψ −1 (c)

=

1 ξ(c + η(d, c)) − α 2 2η (d, c)



ψ −1 (c+η(d,c))

ψ −1 (c)

(ψ(v) − c)α (ξ  o ψ)(v)ψ  (v)dv.

It follows that ξ(c) + ξ(c + η(d, c)) − M 1 − M2 2  ψ−1 (c+η(d,c)) 1 = α [(ψ(v) − c)α − (c + η(d, c) − ψ(v))α ] 2η (d, c) ψ−1 (c) × (ξ  o ψ)(v)ψ  (v)dv.

(4.11)

Next, we prove the second pair equality of the lemma.  η(d, c) 1 Let M3 = ((1 − δ)α − δ α )ξ  (c + (1 − δ)η(d, c))dδ 2 0  1 η(d, c) = (1 − δ)α ξ  (c + (1 − δ)η(d, c))dδ 2 0   1 α  − δ ξ (c + (1 − δ)η(d, c))dδ 0

 1 ξ(c) + ξ(c + η(d, c)) α − = δ α−1 ξ(c + (1 − δ)η(d, c))dδ 2 2 0   1 + δ α−1 ξ(c + δη(d, c))dδ 0

ξ(c) + ξ(c + η(d, c)) = 2   ψ−1 (c+η(d,c)) α − α (c + η(d, c) − ψ(v))α−1 (ξ o ψ)(v)ψ  (v)dv 2η (d, c) ψ−1 (c)   ψ−1 (c+η(d,c)) α−1  + (ψ(v) − c) (ξ o ψ)(v)ψ (v)dv ψ −1 (c)

100

Integral Inequalities and Generalized Convexity =

ξ(c) + ξ(c + η(d, c)) Γ(α + 1)  α:ψ − α J −1 + (ξ o ψ)ψ −1 (c + η(d, c)) 2 2η (d, c) ψ (c)  −1 (c) . (4.12) + Jψα:ψ −1 (c+η(d,c))− (ξ o ψ)ψ

This completes the proof.

Remark 4.3.2. When η(d, c) = d − c, then above lemma reduces to Lemma 3.1 of [98], i.e. Γ(α + 1)  α:ψ ξ(c) + ξ(d) − J −1 + (ξ o ψ)(ψ −1 (d)) 2 2(d − c)α ψ (c)  −1 (c)) + Jψα:ψ −1 (d)− (ξ o ψ)(ψ  ψ−1 (d) 1 = [(ψ(v) − c)α − (d − ψ(v))α ](ξ  o ψ)(v)ψ  (v)dv 2(d − c)α ψ−1 (c)  d−c 1 = ((1 − δ)α − δ α )ξ  (c + (1 − δ)(d − c))dδ. 2 0 Lemma 4.3.2. Let K ⊆ R be an open invex subset with respect to η : K×K → R and c, d ∈ K with c < c + η(d, c). Suppose that ξ : K → R is a differentiable function. If ξ  is preinvex function on K and ξ  ∈ L1 [c, c + η(d, c)], ψ(x) is an increasing and positive monotone function on (c, c + η(d, c)), having a continuous derivative ψ  (x) on (c, c + η(d, c)) and α ∈ (0, 1). Then,  Γ(α + 1)  α:ψ −1 Jψ−1 (c)+ (ξ o ψ)ψ −1 (c + η(d, c)) + Jψα:ψ (c) −1 (c+η(d,c))− (ξ o ψ)ψ α 2η (d, c)   ψ−1 (c+η(d,c))  1 k(ξ  oψ)(v)ψ  (v)dv − ξ c + η(d, c) = 2 ψ −1 (c)  ψ−1 (c+η(d,c)) 1 + α [(c + η(d, c) − ψ(v))α − (ψ(v) − c)α ] 2η (d, c) ψ−1 (c) × (ξ  o ψ)(v)ψ  (v)dv, where k= Proof Let



1 2, − 12 ,

  ψ −1 c + 12 η(d, c) ≤ v ≤ ψ −1 (c + η(d, c)),   ψ −1 (c) < v < ψ −1 c + 12 η(d, c) .

N1 = −

1 2



ψ −1 (c+ 12 η(d,c))

(ξ  o ψ)(v)ψ  (v)dv

ψ −1 (c)

  1 1 1 = ξ(c) − ξ c + η(d, c) , 2 2 2

(4.13)

Integral Inequalities for Generalized Preinvex Functions 1 2

N2 =



ψ

−1

(c+η(d,c))

ψ −1 (c+ 12 η(d,c))

(ξ  o ψ)(v)ψ  (v)dv

  1 1 1 = ξ (c + η(d, c)) − ξ c + η(d, c) , 2 2 2 N3 =

1 2η α (d, c)



ψ −1 (c+η(d,c))

ψ −1 (c)

1 α = − ξ(c) + α 2 2η (d, c)



101

(4.14)

(c + η(d, c) − ψ(v))α (ξ  o ψ)(v)ψ  (v)dv

ψ −1 (c+η(d,c))

ψ −1 (c)

(c + η(d, c) − ψ(v))α−1 (ξ o ψ)(v)ψ  (v)dv

Γ(α + 1) α:ψ 1 J −1 + (ξ o ψ)ψ −1 (c + η(d, c)), = − ξ(c) + α 2 2η (d, c) ψ (c)

(4.15)

and N4 = −

1 α 2η (d, c)



ψ −1 (c+η(d,c))

ψ −1 (c)

1 α = − ξ(c + η(d, c)) + α 2 2η (d, c)

(ψ(v) − c)α (ξ  o ψ)(v)ψ  (v)dv 

ψ −1 (c+η(d,c))

ψ −1 (c)

(ψ(v) − c)α−1 (ξ o ψ)(v)ψ  (v)dv

Γ(α + 1) α:ψ 1 −1 J −1 (c). = − ξ(c + η(d, c)) + α − (ξ o ψ)ψ 2 2η (d, c) ψ (c+η(d,c))

(4.16)

Adding (4.13), (4.14), (4.15) and (4.16), we have Γ(α + 1)  α:ψ J −1 + (ξ o ψ)ψ −1 (c + η(d, c)) 2η α (d, c) ψ (c)    1 α:ψ −1 + Jψ−1 (c+η(d,c))− (ξ o ψ)ψ (c) − ξ c + η(d, c) . 2

N1 + N2 + N 3 + N4 =

This completes the proof. Remark 4.3.3. When η(d, c) = d − c, then above lemma reduces to Lemma 3.2 of [98], i.e.  Γ(α + 1)  α:ψ α:ψ −1 −1 (ξ o ψ)ψ (d) + J (ξ o ψ)(ψ (c)) J −1 + ψ −1 (d)− 2(d − c)α ψ (c)    ψ−1 (d) c+d −ξ = k(ξ  oψ)(v)ψ  (v)dv 2 ψ −1 (c)  ψ−1 (d) 1 + [(d − ψ(v))α − (ψ(v) − c)α ](ξ  o ψ)(v)ψ  (v)dv. 2(d − c)α ψ−1 (c) Now, we derive some fractional Hermite–Hadamard inequalities using the above results.

102

Integral Inequalities and Generalized Convexity

Theorem 4.3.2. Let K ⊆ R be an open invex subset with respect to η : K × K → R and c, d ∈ K with c < c + η(d, c) such that ξ  ∈ L1 [c, c + η(d, c)]. Suppose that ξ : K → R is a differentiable function. If |ξ  | is preinvex function on K, ψ(x) is an increasing and positive monotone function on (c, c + η(d, c)), having a continuous derivative ψ  (x) on (c, c + η(d, c)) and α ∈ (0, 1). Then,   ξ(c) + ξ(c + η(d, c)) Γ(α + 1)  α:ψ  − α J −1 + (ξ o ψ)ψ −1 (c + η(d, c))  2 2η (d, c) ψ (c)    1 η(d, c)  α:ψ −1 1 − α [|ξ  (c)| + |ξ  (d)|]. + Jψ−1 (c+η(d,c))− (ξ o ψ)ψ (c)  ≤ 2(α + 1) 2 Proof From Lemma 4.3.1 and definition of preinvexity, we have   ξ(c) + ξ(c + η(d, c)) Γ(α + 1)  α:ψ  − α J −1 + (ξ o ψ)ψ −1 (c + η(d, c))  2 2η (d, c) ψ (c)   −1 + Jψα:ψ (ξ o ψ)ψ (c)  −1 (c+η(d,c))−   ψ−1 (c+η(d,c))  1  [(ψ(v) − c)α − (c + η(d, c) − ψ(v))α ] = α  2η (d, c) ψ−1 (c)

× (ξ  o ψ)(v)ψ  (v)dv|  η(d, c) 1 ≤ |(1 − δ)α − δ α ||ξ  (c + (1 − δ)η(d, c))|dδ 2 0  η(d, c) 1 ≤ |(1 − δ)α − δ α |[δ|ξ  (c)| + (1 − δ)|ξ  (d)|]dδ 2 0   1/2  1 1/2 η(d, c)  = |ξ (c)| δ(1 − δ)α dδ − δ α+1 dδ + δ α+1 dδ 2 0 0 1/2    1  1/2 1/2 α  − δ(1 − δ) dδ + |ξ (d)| (1 − δ)α+1 dδ − (1 − δ)δ α dδ 1/2

+



0

1 1/2

(1 − δ)δ α dδ −

η(d, c) = 2(α + 1)



1 1/2

(1 − δ)α+1 dδ

  1 1 − α [|ξ  (c)| + |ξ  (d)|]. 2



0

This completes the proof. Remark 4.3.4. When η(d, c) = d−c, then above theorem reduces to Theorem 3.4 of [98], i.e.   ξ(c) + ξ(d) Γ(α + 1)  α:ψ  J −1 + (ξ o ψ)ψ −1 (d) −  2 2(d − c)α ψ (c)    1 d−c  α:ψ −1 + Jψ−1 (d)− (ξ o ψ)ψ (c)  ≤ 1 − α [|ξ  (c)| + |ξ  (d)|]. 2(α + 1) 2

Integral Inequalities for Generalized Preinvex Functions

103 d−2c 2 .

Example 4.3.1. Let c = 0, d = 2, ξ(x) = x, ψ(x) = x and η(d, c) = Then all the assumptions of above theorem are satisfied. Clearly, ξ(c) + ξ(c + ξ(c) + ξ(c + η(d, c)) = 2 2

d−2c 2 )

=

1 , 2

(4.17)

and  Γ(α + 1)  α:ψ α:ψ −1 −1 J (ξ o ψ)ψ (c + η(d, c)) + J (ξ o ψ)ψ (c) −1 + −1 − ψ (c+η(d,c)) 2η α (d, c) ψ (c)  1     1 ΓαΓ2 1 1 α α + = . (1 − v)α−1 vdv + v α−1 vdv = = 2 0 2 Γ(α + 2) α + 1 2 0 (4.18) From (4.17) and (4.18), we get   ξ(c) + ξ(c + η(d, c)) Γ(α + 1)  α:ψ  − α J −1 + (ξ o ψ)ψ −1 (c + η(d, c))  2 2η (d, c) ψ (c)   −1 + Jψα:ψ (c)  = 0 −1 (c+η(d,c))− (ξ o ψ)ψ Next,

η(d, c) 2(α + 1)



1 1− α 2



  1 d − 2c [|ξ (c)| + |ξ (d)|] = 1− α 2(α + 1) 2   1 1 = 1 − α > 0. (α + 1) 2 



Theorem 4.3.3. Let K ⊆ R be an open invex subset with respect to η : K × K → R and c, d ∈ K with c < c + η(d, c) such that ξ  ∈ L1 [c, c + η(d, c)]. Suppose that ξ : K → R is a differentiable function. If |ξ  | is preinvex function on K, ψ(x) is an increasing and positive monotone function on (c, c + η(d, c)), having a continuous derivative ψ  (x) on (c, c + η(d, c)) and α ∈ (0, 1). Then,    Γ(α + 1)  α:ψ α:ψ −1 −1  J (ξ o ψ)ψ (c + η(d, c)) + J (ξ o ψ)ψ (c) −1 + −1 − ψ (c+η(d,c))  2η α (d, c) ψ (c)      |ξ(d) − ξ(c)| 1 η(d, c) 1  − ξ c + η(d, c)  ≤ + 1 − α [|ξ  (c)| + |ξ  (d)|]. 2 2 2(α + 1) 2

Proof Using Lemma 4.3.2, we have    Γ(α + 1)  α:ψ α:ψ −1 −1   2η α (d, c) Jψ−1 (c)+ (ξ o ψ)ψ (c + η(d, c)) + Jψ−1 (c+η(d,c))− (ξ o ψ)ψ (c)    ψ−1 (c+η(d,c))     1    k(ξ o ψ)(v)ψ (v)dv  − ξ c + η(d, c)  ≤   ψ−1 (c)  2  −1   ψ (c+η(d,c)) 1  + α [(c + η(d, c) − ψ(v))α − (ψ(v) − c)α ]  2η (d, c) ψ−1 (c) × (ξ  o ψ)(v)ψ  (v)dv| .

104

Integral Inequalities and Generalized Convexity

Using previous theorem and definition of preinvexity, we have    Γ(α + 1)  α:ψ α:ψ −1 −1  J (ξ o ψ)ψ (c + η(d, c)) + J (ξ o ψ)ψ (c) −1 + −1 − ψ (c+η(d,c))  2η α (d, c) ψ (c)      |ξ(d) − ξ(c)| η(d, c) 1 1  + 1 − α [|ξ  (c)| + |ξ  (d)|]. − ξ c + η(d, c)  ≤ 2 2 2(α + 1) 2 This completes the proof.

Remark 4.3.5. When η(d, c) = d−c, then above theorem reduces to Theorem 3.5 of [98], i.e.    Γ(α + 1)  α:ψ α:ψ −1 −1  (ξ o ψ)(ψ (d)) + J (ξ o ψ)(ψ (c)) J ψ −1 (d)−  2(d − c)α ψ−1 (c)+     c + d  |ξ(d) − ξ(c)| (d − c) 1 −ξ ≤ + 1 − [|ξ  (c)| + |ξ  (d)|].  2 2 2(α + 1) 2α

Example 4.3.2. Let c = 0, d = 1, ξ(x) = x/2, ψ(x) = x and η(d, c) = d − 3c. Then all the assumptions of above theorem are satisfied. Clearly,  Γ(α + 1)  α:ψ α:ψ −1 −1 J (ξ o ψ)ψ (c + η(d, c)) + J (ξ o ψ)ψ (c) −1 + −1 − ψ (c+η(d,c)) 2η α (d, c) ψ (c)  1     1 1 1 α α ΓαΓ2 α−1 α−1 + = , (1 − v) vdv + v vdv = = 4 0 4 Γ(α + 2) α + 1 4 0 (4.19) and     1 1 1 ξ c + η(d, c) = ξ c + (d − 3c) = . 2 2 4

(4.20)

From (4.19) and (4.20), we get    Γ(α + 1)  α:ψ α:ψ −1 −1  J (ξ o ψ)ψ (c + η(d, c)) + J (ξ o ψ)ψ (c) −1 + −1 − ψ (c+η(d,c))  2η α (d, c) ψ (c)    1 (4.21) − ξ c + η(d, c)  = 0. 2 Next,

  η(d, c) 1 |ξ(d) − ξ(c)| + 1 − α [|ξ  (c)| + |ξ  (d)|] 2 2(α + 1) 2   1 1 1 1 − α > 0. = + 4 2(α + 1) 2

Theorem 4.3.4. Let K ⊆ R be an open invex subset with respect to η : K × K → R and c, d ∈ K with c < c + η(d, c) such that ξ  ∈ L1 [c, c + η(d, c)].

Integral Inequalities for Generalized Preinvex Functions

105

Suppose that ξ : K → R is a differentiable function. If |ξ  |q is preinvex function on K for some fixed q > 1, ψ(x) is an increasing and positive monotone function on (c, c + η(d, c)), having a continuous derivative ψ  (x) on (c, c + η(d, c)) and α ∈ (0, 1). Then,   ξ(c) + ξ(c + η(d, c)) Γ(α + 1)  α:ψ  − α J −1 + (ξ o ψ)ψ −1 (c + η(d, c))  2 2η (d, c) ψ (c) 1    η(d, c) |ξ (c)|q + |ξ  (d)|q q  α:ψ −1 , + Jψ−1 (c+η(d,c))− (ξ o ψ)ψ (c)  ≤ 1 2 2(αp + 1) p where

1 p

+

1 q

= 1.

Proof From Lemma 4.3.1, we have   ξ(c) + ξ(c + η(d, c)) Γ(α + 1)  α:ψ  − α J −1 + (ξ o ψ)ψ −1 (c + η(d, c))  2 2η (d, c) ψ (c)   −1 + Jψα:ψ (c)  −1 (c+η(d,c))− (ξ o ψ)ψ  η(d, c) 1 α |δ − (1 − δ)α ||ξ  (c + δη(d, c))|dδ. ≤ 2 0 Using H¨ older’s inequality and definition of preinvexity, we have   ξ(c) + ξ(c + η(d, c)) Γ(α + 1)  α:ψ  − α J −1 + (ξ o ψ)ψ −1 (c + η(d, c))  2 2η (d, c) ψ (c)   −1 + Jψα:ψ (c)  −1 (c+η(d,c))− (ξ o ψ)ψ  1  p1  1  q1 η(d, c) α α p  q |δ − (1 − δ) | dδ |ξ (c + δη(d, c))| dδ ≤ 2 0 0  1  p1  1  q1 η(d, c) αp  q  q ≤ |1 − 2δ| dδ ((1 − δ)|ξ (c)| + δ|ξ (d)| ) 2 0 0   1 η(d, c) |ξ (c)|q + |ξ  (d)|q q = . 1 2 2(αp + 1) p This completes the proof. Corollary 4.3.1. If η(d, c) = d − c, then we have the following result:   ξ(c) + ξ(d) Γ(α + 1)  α:ψ  − J −1 + (ξ o ψ)ψ −1 (d)  2 2(d − c)α ψ (c)   1  (d − c) |ξ (c)|q + |ξ  (d)|q q  α:ψ −1 . + Jψ−1 (d)− (ξ o ψ)ψ (c)  ≤ 1 2 2(αp + 1) p

106

Integral Inequalities and Generalized Convexity

Theorem 4.3.5. Let K ⊆ R be an open invex subset with respect to η : K × K → R and c, d ∈ K with c < c + η(d, c) such that ξ  ∈ L1 [c, c + η(d, c)]. Suppose that ξ : K → R is a differentiable function. If |ξ  |q is preinvex function on K for some fixed q > 1, ψ(x) is an increasing and positive monotone function on (c, c + η(d, c)), having a continuous derivative ψ  (x) on (c, c + η(d, c)) and α ∈ (0, 1). Then, Proof Using Lemma 4.3.1, H¨ older’s inequality and definition of preinvexity, we have   ξ(c) + ξ(c + η(d, c)) Γ(α + 1)  α:ψ  − α J −1 + (ξ o ψ)ψ −1 (c + η(d, c))  2 2η (d, c) ψ (c)  p1  η(d, c)  1  −1 α α (ξ o ψ)ψ (c) ≤ |δ − (1 − δ) |dδ + Jψα:ψ  −1 (c+η(d,c))− 2 0  1  q1 × |δ α − (1 − δ)α ||ξ  (c + δη(d, c))|q dδ 0

  p1  1 1/2 η(d, c) α α α α = ((1 − δ) − δ )dδ + (δ − (1 − δ) )dδ 2 0 1/2  1/2 × ((1 − δ)α − δ α )((1 − δ)|ξ  (c)|q + δ|ξ  (d)|q )dδ 0

+



1 α

1/2

α





q

q

(δ − (1 − δ) )((1 − δ)|ξ (c)| + δ|ξ (d)| )dδ

η(d, c) = (α + 1)



1 1− α 2



|ξ  (c)|q + |ξ  (d)|q 2

 q1

 q1

.

Corollary 4.3.2. If η(d, c) = d − c, then we have the following result:   ξ(c) + ξ(d) Γ(α + 1)  α:ψ  J −1 + (ξ o ψ)ψ −1 (d) −  2 2(d − c)α ψ (c)  1    1 d−c |ξ (c)|q + |ξ  (d)|q q  −1 1 − (ξ o ψ)ψ (c) ≤ . + Jψα:ψ  −1 (d)− (α + 1) 2α 2

4.4

Application to Special Means

We shall start with the following proposition [158]. Here, A(c, d), Ln (c, d) and H(c, d) are Arithmetic mean, Generalized logarithmic mean and Harmonic mean, respectively, which are defined in Section 2.3.1 of Chapter 2.

Integral Inequalities for Generalized Preinvex Functions

107

Proposition 4.4.1. Let c, c + η(d, c) ∈ R+ , c < c + η(d, c). Then n

n

|A(c , (c + η(d, c)) ) −

Lnn (c, c

+ η(d, c))| ≤



nη(d, c) 1

2(p + 1) p

c(n−1)q + d(n−1)q 2

 q1

.

Proof Applying Theorem 4.3.4 with ξ(x) = xn , ψ(x) = x, α = 1. Then we compute the result easily. Proposition 4.4.2. Let c, c + η(d, c) ∈ R+ , c < c + η(d, c). Then c

|A(e , e

c+η(d,c)

c

) − L(e , e

c+η(d,c)

)| ≤

η(d, c) 1

2(p + 1) p



ecq + edq 2

 q1

.

Proof Applying Theorem 4.3.4 with ξ(x) = ex , ψ(x) = x, α = 1. Then we compute the result easily. Proposition 4.4.3. Let c, c + η(d, c) ∈ R+ , c < c + η(d, c). Then   −1 H (c, c + η(d, c)) − L−1 (c, c + η(d, c)) ≤

   q1 1 1 1 + . 1 2q d2q 2(p + 1) p 2 c

Proof Applying Theorem 4.3.4 with ξ(x) = compute the result easily.

η(d, c)

1 x , ψ(x)

= x, α = 1. Then we

Proposition 4.4.4. Let c, c + η(d, c) ∈ R+ , c < c + η(d, c). Then n

n

|A(c , (c + η(d, c)) ) −

Lnn (c, c

nη(d, c) + η(d, c))| ≤ 4



c(n−1)q + d(n−1)q 2

 q1

.

Proof Applying Theorem 4.3.5 with ξ(x) = xn , ψ(x) = x, α = 1. Then we compute the result easily. Proposition 4.4.5. Let c, c + η(d, c) ∈ R+ , c < c + η(d, c). Then |A(ec , ec+η(d,c) ) − L(ec , ec+η(d,c) )| ≤

η(d, c) 4



ecq + edq 2

 q1

.

Proof Applying Theorem 4.3.5 with ξ(x) = ex , ψ(x) = x, α = 1. Then we compute the result easily. Proposition 4.4.6. Let c, c + η(d, c) ∈ R+ , c < c + η(d, c). Then

   q1   −1 1 1 H (c, c + η(d, c)) − L−1 (c, c + η(d, c)) ≤ η(d, c) 1 + 2q . 4 2 c2q d

Proof Applying Theorem 4.3.5 with ξ(x) = compute the result easily.

1 x , ψ(x)

= x, α = 1. Then we

108

Integral Inequalities and Generalized Convexity

4.5

Generalized (m,h)-Preunivex Mappings via k -Fractional Integrals

We present the definition of (m, h)-preunivex function [157]. Definition 4.5.1. Let K ⊆ R be an open m-invex subset with respect to η : K × K × (0, 1] → R and let h : [0, 1] → R\{0}. A function ξ : K → R is said to be generalized (m, h)-preunivex with respect to η, φ and b if ξ(mx + δη(y, x, m)) ≤ mξ(x) + h(δ)b(y, x, h(δ))φ[ξ(y) − mξ(x)]. Remark 4.5.1. Now, we will discuss some special cases of Definition 4.5.1, we have (1) choosing h(δ) = 1, we obtain the definition of generalized (m, p)preunivex functions; (2) choosing h(δ) = δ s for s ∈ (0, 1], we obtain the definition of generalized (m, s)-Breckner-preunivex functions; (3) choosing h(δ) = δ −s for s ∈ (0, 1], we obtain the definition of generalized (m, s)-Godunova-Levin-Dragomir-preunivex functions; (4) choosing h(δ) = δ(1−δ), we obtain the definition of generalized (m, tgs)preunivex functions; √

δ , we obtain the definition of generalized m-M T (5) choosing h(δ) = 2√1−δ preunivex functions.

By using Lemma 4.2.1, we prove the following theorem [157]. Theorem 4.5.1. Let K ⊆ R be an open m-invex subset with respect to η : K × K × (0, 1] → R\{0} for some fixed m ∈ (0, 1], and let c, d ∈ K, c < d with η(d, c, m) > 0. Assume that ξ : K → R is a twice differentiable function   on K such that ξ is integrable on [mc, mc + η(d, c, m)]. If |ξ |q for q ≥ 1 is a generalized (m, h)-preunivex function with respect to η, φ and b and h : [0, 1] → R\{0}, then the following inequality for k-fractional integrals with x ∈ [c, d], λ ∈ [0, 1], α > 0, k > 0 exists:   α +2  η k (x, c, m)  1− q1   |Iξ,η (α, k; x, λ, m, c, d)| ≤ A0 (k, α, λ)  α ( k + 1)η(d, c, m)   q1    × mA0 (k, α, λ)|ξ  (c)|q + φ(|ξ (x)|q − m|ξ (c)|q )A1 (k, α, λ, c, x; h)    (−1) αk +2 η αk +2 (x, d, m)   +  ( αk + 1)η(d, c, m)   q1      q q q , × mA0 (k, α, λ)|ξ (d)| + φ(|ξ (x)| − m|ξ (d)| )A2 (k, α, λ, d, x; h)

Integral Inequalities for Generalized Preinvex Functions   1  α where A0 (k, α, λ) = 0 δ  αk + 1 λ − δ k  dδ  2k  αk [( αk +1)λ]1+ α − ( αk +1)λ + 1 , 0 ≤ λ ≤ 1 , α α α +2 2 k k +2 k +1 = ( α +1)λ 1  k − α1 , < λ ≤ 1, α 2

A1 (k, α, λ, c, x; h) = and A2 (k, α, λ, d, x; h) =

k





1

0

1 0

+2

k

109

+1

  α  α  + 1 λ − δ k  h(δ)b(x, c, h(δ))dδ, 0 ≤ λ ≤ 1, δ k

  α  α  + 1 λ − δ k  h(δ)b(x, d, h(δ))dδ, 0 ≤ λ ≤ 1. δ k

Proof For the proof of this theorem, we will apply Lemma 4.2.1 and the Power mean inequality, we have  1   α +2  1− q1   η k (x, c, m)  α  α   + 1 λ − δ k  dδ δ |Iξ,η (α, k; x, λ, m, c, d)| ≤  α ( k + 1)η(d, c, m)  0 k  1   q1    α  α q + 1 λ − δ k  |ξ (mc + δη(x, c, m))| dδ × δ k 0   1    1− q1   (−1) αk +2 η αk +2 (x, d, m)  α α   + 1 λ − δ k  dδ +  δ ( αk + 1)η(d, c, m)  0 k  1   q1    α  α q k + 1 λ − δ  |ξ (md + δη(x, d, m))| dδ × δ k 0 1− 1

= A0 q (k, α, λ)  α  1   q1     η k +2 (x, c, m)  α  α q   + 1 λ − δ k  |ξ (mc + δη(x, c, m))| dδ δ ×  α ( k + 1)η(d, c, m)  0 k    (−1) αk +2 η αk +2 (x, d, m)    + ( αk + 1)η(d, c, m)   1   q1     α  α q × + 1 λ − δ k  |ξ (md + δη(x, d, m))| dδ δ . (4.22) k 0 

Since |ξ (x)|q is generalized (m, h)-preunivex on [mc, mc + η(d, c, m)], we get  1     α  α + 1 λ − δ k  |ξ (mc + δη(x, c, m))|q dδ δ k 0  1     α  α ≤ δ + 1 λ − δ k  (m|ξ  (c)|q + h(δ)b(x, c, h(δ)) k 0    × φ(|ξ (x)|q − m|ξ (c)|q )) dt 



= mA0 (k, α, λ)|ξ  (c)|q + φ(|ξ (x)|q − m|ξ (c)|q )A1 (k, α, λ, c, x; h) (4.23)

110

Integral Inequalities and Generalized Convexity

and



   α  α  + 1 λ − δ k  |ξ (md + δη(x, d, m))|q dδ δ k 0  1     α  α ≤ δ + 1 λ − δ k  (m|ξ  (d)|q + h(δ)b(x, d, h(δ)) k 0    × φ(|ξ (x)|q − m|ξ (d)|q )) dδ 1





= mA0 (k, α, λ)|ξ  (d)|q + φ(|ξ (x)|q − m|ξ (d)|q )A2 (k, α, λ, d, x; h). (4.24)

Hence, if we use (4.23) and (4.24) in (4.22), then we have the desired result. This completes the proof. Now, we will discuss some special cases of Theorem 4.5.1 [157]. 

Corollary 4.5.1. In Theorem 4.5.1, if |ξ |q for q ≥ 1 is generalized (m,s)Breckner-preunivex functions, then for s ∈ (0, 1] and m ∈ (0, 1], we have   α +2  η k (x, c, m)  1− q1   |Iξ,η (α, k; x, λ, m, c, d)| ≤ A0 (k, α, λ)  α ( k + 1)η(d, c, m)  



1

× (mA0 (k, α, λ)|ξ  (c)|q + φ(|ξ (x)|q − m|ξ (c)|q )B1 (k, α; λ, s, c, x)) q    (−1) αk +2 η αk +2 (x, d, m)    + ( α + 1)η(d, c, m)  k

   1 × (mA0 (k, α, λ)|ξ  (d)|q + φ(|ξ (x)|q − m|ξ (d)|q )B2 (k, α; λ, s, d, x)) q ,

where

B1 (k, α; λ, s, c, x) = and B2 (k, α; λ, s, d, x) =

 

1 0

1 0

 α   α  + 1 λ − δ k  b(x, c, δ s )dδ δ s+1  k

 α   α  + 1 λ − δ k  b(x, d, δ s )dδ. δ s+1  k

Corollary 4.5.2. In Corollary 4.5.1, if φ : R → R is defined by φ(c) = c and b(., ., .) ≡ 1, then we have   α +2  η k (x, c, m)  1− q1   |Iξ,η (α, k; x, λ, m, c, d)| ≤ A0 (k, α, λ)  α ( k + 1)η(d, c, m)   q1    × mA0 (k, α, λ)|ξ  (c)|q + (|ξ (x)|q − m|ξ (c)|q )H1 (k, α, λ; s)    (−1) αk +2 η αk +2 (x, d, m)   +  ( αk + 1)η(d, c, m)   q1      q q q , × mA0 (k, α, λ)|ξ (d)| + (|ξ (x)| − m|ξ (d)| )H1 (k, α, λ; s)

Integral Inequalities for Generalized Preinvex Functions where   1 α H1 (k, α, λ; s) = 0 δ s+1  αk + 1 λ − δ k  dδ =

 k(s+2) 1+ α α  2α k [( k +1)λ] (s+2)( α k +s+2)  ( αk +1)λ − α 1 , s+2 k +s+2

−

(α k +1)λ s+2

+

1

α k +s+2

,

0≤λ≤ 1 α k +1

111

1

α k +1

,

< λ ≤ 1.

Corollary 4.5.3. In Corollary 4.5.2, if the mapping η(d, c, m) = d−mc along with m = 1, taking x = c+d 2 , then for s ∈ (0, 1], we have    2 1  2 αk −1  c+d   ≤ (d − c) A1− q (k, α, λ) , λ, 1, c, d I α, k; α 0 α  (d − c) k −1 ξ  2 8( k + 1)      q1    c + d q  − |ξ  (c)|q H1 (k, α, λ; s) × A0 (k, α, λ)|ξ  (c)|q + ξ  2       q1    c + d q   q q   . + A0 (k, α, λ)|ξ (d)| + ξ  − |ξ (d)| H1 (k, α, λ; s) 2

Remark 4.5.2. In Corollary 4.5.3,

(1) if λ = 0, then we have     α   k −1 Γ (α + k) k α α ξ c + d − 2 [k J( c+d )− ξ(c) + k J( c+d )+ ξ(d)] α  k 2 2 2 (d − c)  1− q1 2 (d − c) 1 ≤ α 8( k + 1) αk + 2      q1   c + d q  1 1  q q   |ξ (c)| + α × α ξ  − |ξ (c)| 2 k +2 k +s+2  q1    q      1 c + d 1 ξ  − |ξ (d)|q , |ξ  (d)|q + α + α   2 k +2 k +s+2

(2) if we choose k = α = 1 and λ = 0, then we get      d   c+d 1   − ξ(x)dx ξ   2 d−c c 1        q1 1− q   c + d q  1 1  (d − c)2 1 q q   |ξ (c)| + ξ ≤  − |ξ (c)| 16 3 3 s+3  2     q1     c + d q  1 1  q q ξ  − |ξ (d)| |ξ (d)| + , +  3 s+3  2

112

Integral Inequalities and Generalized Convexity

(3) if λ = 1, then we have    ξ(c) + ξ(d) 2 αk −1 Γk (α + k)  α α   − [ J ξ(c) + J ξ(d)] α k ( c+d )− k ( c+d )+   2 2 2 (d − c) k 1  1− q  α α  (d − c)2 αk ( αk + 3) k ( k + 3) ≤ α |ξ  (c)|q 8( k + 1) 2( αk + 2) 2( αk + 2)     q1 α α   c + d q ( + s + 3) ξ  − |ξ  (c)|q + αk k  ( k + s + 2)(s + 2)  2   α α k ( k + 3) |ξ  (d)|q + 2( αk + 2)     q1  α α   c + d q  q k ( k + s + 3)   ξ + α ,  − |ξ (d)| ( k + s + 2)(s + 2)  2

(4) if we choose k = α = λ = 1, then we get    d  (d − c)2  2 1− q1  ξ(c) + ξ(d) 1   − ξ(x)dx ≤    2 d−c c 16 3      q1   c + d q  2  (s + 4) q q   |ξ (c)| + ξ ×  − |ξ (c)| 3 (s + 2)(s + 3)  2      q1    c + d q  2  (s + 4) q q   |ξ (d)| + ξ + ,  − |ξ (d)| 3 (s + 2)(s + 3)  2

(5) if λ = 13 , then we have      1  ξ(c) + 4ξ c + d + ξ(d) 6 2

 α  2 k −1 Γk (α + k) [k J(αc+d )− ξ(c) + k J(αc+d )+ ξ(d)] α 2 2 (d − c) k      2 1 (d − c) 1− q 1 1 ≤ α k, α, A0 A0 k, α, |ξ  (c)|q 8( k + 1) 3 3   q1       c + d q  − |ξ  (c)|q H1 k, α, 1 , s + ξ  2 3    1 |ξ  (d)|q + A0 k, α, 3   q1        c + d q  1 q  − |ξ (d)| H1 k, α, , s , + ξ  2 3 −

Integral Inequalities for Generalized Preinvex Functions

113

(6) if we choose k = α = 1 and λ = 13 , then we get        d  (d − c)2  8 1− q1 1 c+d 1   + ξ(d) − ξ(x)dx ≤  ξ(c) + 4ξ  6 2 d−c c 16 81    q1       c + d q  1 8  q q   |ξ (c)| + ξ ×  − |ξ (c)| H1 1, 1, 3 , s 81 2   q1         c + d q  1 8  q q   |ξ (d)| + ξ + ,  − |ξ (d)| H1 1, 1, 3 , s 81 2 (7) if λ = 12 , then we have      1  ξ(c) + 2ξ c + d + ξ(d) 4 2

 α  2 k −1 Γk (α + k) [k J(αc+d )− ξ(c) + k J(αc+d )+ ξ(d)] α 2 2 (d − c) k   2 (d − c) 1− q1 1 ≤ α A0 k, α, 8( k + 1) 2      q1       c + d q  1 1  q q   |ξ (c)| + ξ × A0 k, α,  − |ξ (c)| H1 k, α, 2 , s 2 2      q1        c + d q  1 1  q q  − |ξ (d)| H1 k, α, , s |ξ (d)| + ξ + A0 k, α, ,  2 2 2 −

(8) if we choose k = α = 1 and λ = 12 , then we get        d  (d − c)2  1 1− q1 1 1 c+d   ξ(c) + 2ξ + ξ(d) − ξ(x)dx ≤   4 2 d−c c 16 6    q1       c + d q  1  1 q q   |ξ (c)| + ξ ×  − |ξ (c)| H1 1, 1, 2 , s 6 2    q1        c + d q  1 1  q q  − |ξ (d)| H1 1, 1, , s |ξ (d)| + ξ + .  6 2 2

114

Integral Inequalities and Generalized Convexity 

Corollary 4.5.4. In Theorem 4.5.1, if |ξ |q for q ≥ 1 is generalized (m,tgs)preunivex functions, then for m ∈ (0, 1], we have   α +2  η k (x, c, m)  1− 1  |Iξ,η (α, k; x, λ, m, c, d)| ≤ A0 q (k, α, λ)  α ( k + 1)η(d, c, m)   q1    × mA0 (k, α, λ)|ξ  (c)|q + φ(|ξ (x)|q − m|ξ (c)|q )E1 (k, α, λ, δ, c, x)    (−1) αk +2 η αk +2 (x, d, m)    + ( αk + 1)η(d, c, m)   q1     , × mA0 (k, α, λ)|ξ  (d)|q + φ(|ξ (x)|q − m|ξ (d)|q )E2 (k, α, λ, δ, d, x) where E1 (k, α, λ, δ, c, x) =



1

  α  α  + 1 λ − δ k  b(x, c, δ(1 − δ))dδ δ 2 (1 − δ)  k

1

  α  α  + 1 λ − δ k  b(x, d, δ(1 − δ))dδ. δ 2 (1 − δ)  k

0

and E2 (k, α, λ, δ, d, x) =



0

Corollary 4.5.5. In Corollary 4.5.4, if φ : R → R is defined by φ(c) = c and b(., ., .) ≡ 1, then we have   α +2  η k (x, c, m)  1− q1   |Iξ,η (α, k; x, λ, m, c, d)| ≤ A0 (k, α, λ)  α ( k + 1)η(d, c, m)   q1    × mA0 (k, α, λ)|ξ  (c)|q + (|ξ (x)|q − m|ξ (c)|q )N1 (k, α, λ)    (−1) αk +2 η αk +2 (x, d, m)    + ( αk + 1)η(d, c, m)   q1     , × mA0 (k, α, λ)|ξ  (d)|q + (|ξ (x)|q − m|ξ (d)|q )N1 (k, α, λ) where   1 α N1 (k, α, λ) = 0 δ 2 (1 − δ)  αk + 1 λ − δ k  dδ  1+ 3k 1+ 4k α α α α α ( α +1)λ  2α k [( k +1)λ] k [( k +1)λ] − − k 12 + α α 3( +3) 2( +4) k = ( α +1)λk 1  k − ( α +3)( α 12 +4) , k

k

1 , α (α k +3)( k +4)

0≤λ≤ 1 α k +1

1

α k +1

,

< λ ≤ 1.

Integral Inequalities for Generalized Preinvex Functions

115

Corollary 4.5.6. In Corollary 4.5.5, if the mapping η(d, c, m) = d−mc along with m = 1, taking x = c+d 2 , then we have    2 1  2 αk −1  c+d   ≤ (d − c) A1− q (k, α, λ) , λ, 1, c, d I α, k; 0 α  (d − c) αk −1 ξ  2 8( k + 1) 

    q1    c + d q  q   × A0 (k, α, λ)|ξ (c)| + ξ  − |ξ (c)| N1 (k, α, λ) 2  q     q1      c + d  − |ξ (d)|q N1 (k, α, λ) . + A0 (k, α, λ)|ξ  (d)|q + ξ  2 

q

Remark 4.5.3. In Corollary 4.5.6,

(1) if λ = 0, then we have     α   k −1 Γ (α + k) k α α ξ c + d − 2  [ J ξ(c) + J ξ(d)] α k ( c+d )− k ( c+d )+   2 2 2 (d − c) k  1− q1 (d − c)2 1 ≤ α 8( k + 1) αk + 2      q1   c + d q 1 1  q ξ  − |ξ  (c)|q |ξ × (c)| + α  ( αk + 3)( αk + 4)  2 k +2  q1        c + d q  1 1  q q   |ξ (d)| + α ξ + α ,  − |ξ (d)| ( k + 3)( αk + 4)  2 k +2 (2) if we choose k = α = 1 and λ = 0, then we get      d  (d − c)2  1 1− q1  1 c+d   − ξ(x)dx ≤ ξ   2 d−c c 16 3   q1   q  1   c + d  1  q |ξ (c)|q + ξ × − |ξ (c)|  3 20  2   q   q1   1  1   c + d  q q |ξ (d)| + ξ + ,  − |ξ (d)| 3 20  2

116

Integral Inequalities and Generalized Convexity

(3) if λ = 1, then we have    ξ(c) + ξ(d) 2 αk −1 Γk (α + k)  α α   − [ J ξ(c) + J ξ(d)] α k ( c+d )− k ( c+d )+   2 2 2 (d − c) k 1  1− q  α α  (d − c)2 αk ( αk + 3) k ( k + 3) ≤ α |ξ  (c)|q 8( k + 1) 2( αk + 2) 2( αk + 2)     q1 α α2 α   c + d q  q k ( k2 + 8 k + 19)   ξ +  − |ξ (c)| 12( αk + 3)( αk + 4)  2   α α k ( k + 3) |ξ  (d)|q + 2( αk + 2)   q1   q α α2 α   ( + 8 + 19)   c + d 2 k ξ  − |ξ (d)|q , + k kα  12( k + 3)( αk + 4)  2

(4) if we choose k = α = λ = 1, then we get    d  (d − c)2  2 1− q1  ξ(c) + ξ(d) 1   − ξ(x)dx ≤    2 d−c c 16 3    q  q1  2  7   c + d  q q |ξ (c)| + ξ ×  − |ξ (c)| 3 60  2   q   q1   2  7   c + d  q q |ξ (d)| + ξ + ,  − |ξ (d)| 3 60  2 (5) if λ = 13 , then we have      1  ξ(c) + 4ξ c + d + ξ(d) 6 2

 α  2 k −1 Γk (α + k) α α [k J( c+d )− ξ(c) + k J( c+d )+ ξ(d)] − α 2 2 (d − c) k   (d − c)2 1− q1 1 ≤ α k, α, A 8( k + 1) 0 3      q1       c + d q  1 1  q q   |ξ (c)| + ξ × A0 k, α,  − |ξ (c)| N1 k, α, 3 3 2    q  q1          1 1 c + d  − |ξ (d)|q N1 k, α, |ξ  (d)|q + ξ + A0 k, α, ,  3 2 3

Integral Inequalities for Generalized Preinvex Functions

117

(6) if we choose k = α = λ = 1 and λ = 13 , then we get        d  (d − c)2  8 1− q1 1 c+d 1   + ξ(d) − ξ(x)dx ≤  ξ(c) + 4ξ  6 2 d−c c 16 81    q  q1     c + d  8   − |ξ  (c)|q N1 1, 1, 1 |ξ (c)|q + ξ ×  81 2 3 q  q1           8  c + d 1  − |ξ (d)|q N1 1, 1, |ξ (d)|q + ξ + ,  81 2 3 (7) if λ = 12 , then we have      1  ξ(c) + 2ξ c + d + ξ(d) 4 2 α

2 k −1 Γk (α + k) − [k J(αc+d )− ξ(c) + α 2 (d − c) k

≤

 

α  k J( c+d )+ ξ(d)] 2

  (d − c)2 1− q1 1 A k, α, 8( αk + 1) 0 2      q1       c + d q 1  − |ξ  (c)|q N1 k, α, 1 |ξ  (c)|q + ξ × A0 k, α,  2 2 2    q1          c + d q  1 1  q q  − |ξ (d)| N1 k, α, |ξ (d)| + ξ , + A0 k, α,  2 2 2

(8) if we choose k = α = 1 and λ = 12 , then we get        d  (d − c)2  1 1− q1 1 1 c+d   + ξ(d) − ξ(x)dx ≤  ξ(c) + 2ξ  4 2 d−c c 16 6  q  q1       c + d  1   − |ξ  (c)|q N1 1, 1, 1 |ξ (c)|q + ξ ×  6 2 2 q  q1           c + d 1 1   − |ξ (d)|q N1 1, 1, |ξ (d)|q + ξ + .  6 2 2

118

Integral Inequalities and Generalized Convexity 

Corollary 4.5.7. In Theorem 4.5.1, if |ξ |q for q ≥ 1 is generalized m-MTpreunivex functions, then for m ∈ (0, 1], we have   α +2  η k (x, c, m)  1− 1  |Iξ,η (α, k; x, λ, m, c, d)| ≤ A0 q (k, α, λ)  α ( k + 1)η(d, c, m)   q1    × mA0 (k, α, λ)|ξ  (c)|q + φ(|ξ (x)|q − m|ξ (c)|q )G1 (k, α, λ, δ, c, x)    (−1) αk +2 η αk +2 (x, d, m)    + ( αk + 1)η(d, c, m)   q1     , × mA0 (k, α, λ)|ξ  (d)|q + φ(|ξ (x)|q − m|ξ (d)|q )G2 (k, α, λ, δ, d, x) where G1 (k, α, λ, δ, c, x) =



1 0

and G2 (k, α, λ, δ, d, x) =



1 0

  √ √   α δ δ  α δ √ dδ + 1 λ − δ k  b x, c, √  2 1−δ k 2 1−δ   √ √   α δ δ  α δ k √ dδ. + 1 λ − δ  b x, d, √  2 1−δ k 2 1−δ

Corollary 4.5.8. In Corollary 4.5.7, if φ : R → R is defined by φ(c) = c and b(., ., .) ≡ 1, then we have   α +2  η k (x, c, m)  1− q1   |Iξ,η (α, k; x, λ, m, c, d)| ≤ A0 (k, α, λ)  α ( k + 1)η(d, c, m)   q1    × mA0 (k, α, λ)|ξ  (c)|q + (|ξ (x)|q − m|ξ (c)|q )R1 (k, α, λ)    (−1) αk +2 η αk +2 (x, d, m)   +  ( αk + 1)η(d, c, m)   q1      q q q , × mA0 (k, α, λ)|ξ (d)| + (|ξ (x)| − m|ξ (d)| )R1 (k, α, λ) where 1 R1 (k, α, λ) = 0

√ δ δ √ 2 1−δ

 α    + 1 λ − δ αk  dδ k

  3λπ( α 1 α 5 1 k +1)   2 β( k + 2 , 2 ) − 16   k   +( αk + 1)λβ([( αk + 1)λ] α ; 52 , 12 ) , = k  −β([( αk + 1)λ] α ; αk + 52 , 12 )   α   3λπ( k +1) − 1 β( α + 5 , 1 ), 16

2

k

2 2

0≤λ≤ 1

α k +1

1

α k +1

,

< λ ≤ 1.

Integral Inequalities for Generalized Preinvex Functions

119

Corollary 4.5.9. In Corollary 4.5.8, if the mapping η(d, c, m) = d−mc along with m = 1, taking x = c+d 2 , then we have    2 1  2 αk −1  c+d   ≤ (d − c) A1− q (k, α, λ) , λ, 1, c, d I α, k; 0 α  (d − c) αk −1 ξ  2 8( k + 1)      q1    c + d q  − |ξ  (c)|q R1 (k, α, λ) × A0 (k, α, λ)|ξ  (c)|q + ξ  2       q1    c + d q   q q   + A0 (k, α, λ)|ξ (d)| + ξ .  − |ξ (d)| R1 (k, α, λ) 2

Remark 4.5.4. In Corollary 4.5.9,

(1) if λ = 0, then we have     α   k −1 Γ (α + k) k α α ξ c + d − 2 [k J( c+d )− ξ(c) + k J( c+d )+ ξ(d)] α  2 2 2 (d − c) k  1− q1 (d − c)2 1 ≤ α 8( k + 1) αk + 2   q1        c + d q  1 α 5 1 1  q q ξ  − |ξ (c)| |ξ (c)| + β + , × α   2 k 2 2 2 k +2     q   q1      α 5 1 1 1 c + d ξ  − |ξ (d)|q |ξ  (d)|q + β + , + α ,   2 k 2 2 2 k +2

(2) if we choose k = α = 1 and λ = 0, then we get      d   c+d 1   − ξ(x)dx ξ   2 d−c c   1− q1    q  q1  5π   c + d  1  (d − c)2 1 q |ξ (c)|q + ξ − |ξ (c)| ≤  16 3 3 32  2  q1    q   5π   c + d  1  q q |ξ (d)| + ξ + ,  − |ξ (d)| 3 32  2

120

Integral Inequalities and Generalized Convexity

(3) if λ = 1, then we have    ξ(c) + ξ(d) 2 αk −1 Γk (α + k)  α α   − [ J ξ(c) + J ξ(d)] α k ( c+d )− k ( c+d )+   2 2 2 (d − c) k 1  1− q  α α  (d − c)2 αk ( αk + 3) k ( k + 3) ≤ α |ξ  (c)|q 8( k + 1) 2( αk + 2) 2( αk + 2)       q1    c + d q 3π( αk + 1) 1 α 5 1 ξ  − |ξ  (c)|q − β + , +   16 2 k 2 2 2   α α k ( k + 3) |ξ  (d)|q + 2( αk + 2)  q1          c + d q 3π( αk + 1) 1  α 5 1 ξ  − |ξ (d)|q , − β + , +   16 2 k 2 2 2

(4) if we choose k = α = λ = 1, then we get    d  ξ(c) + ξ(d)  1   − ξ(x)dx    2 d−c c 1      q  q1 1− q  7π   c + d  2  (d − c)2 2 q q |ξ (c)| + ξ ≤  − |ξ (c)| 16 3 3 32  2  q1    q   7π   c + d  2  q q , |ξ (d)| + ξ +  − |ξ (d)| 3 32  2

(5) if λ = 13 , then we have      1  ξ(c) + 4ξ c + d + ξ(d) 6 2

 α  2 k −1 Γk (α + k) α α − [k J( c+d )− ξ(c) + k J( c+d )+ ξ(d)] α k 2 2 (d − c)   (d − c)2 1− q1 1 ≤ α A k, α, 8( k + 1) 0 3        q1     c + d q  1 1  q q   |ξ (c)| + ξ × A0 k, α,  − |ξ (c)| R1 k, α, 3 3 2    q1          c + d q  1 1  q q   |ξ (d)| + ξ + A0 k, α, ,  − |ξ (d)| R1 k, α, 3 3 2

Integral Inequalities for Generalized Preinvex Functions

121

(6) if we choose k = α = 1 and λ = 13 , then we get        d  (d − c)2  8 1− q1 1 c+d 1   + ξ(d) − ξ(x)dx ≤  ξ(c) + 4ξ  6 2 d−c c 16 81  q  q1       c + d  8   − |ξ  (c)|q R1 1, 1, 1 |ξ (c)|q + ξ ×  81 2 3   q  q1         1 c + d 8   − |ξ (d)|q R1 1, 1, , |ξ (d)|q + ξ +  81 2 3

(7) if λ = 12 , then we have      1  ξ(c) + 2ξ c + d + ξ(d) 4 2

 α  2 k −1 Γk (α + k) α α − [k J( c+d )− ξ(c) + k J( c+d )+ ξ(d)] α k 2 2 (d − c)   (d − c)2 1− q1 1 ≤ α k, α, A 8( k + 1) 0 2      q1       c + d q  1 1  q q  − |ξ (c)| R1 k, α, |ξ (c)| + ξ × A0 k, α,  2 2 2  q  q1          c + d  1  − |ξ  (d)|q R1 k, α, 1 |ξ  (d)|q + ξ + A0 k, α, ,  2 2 2

(8) if we choose k = α = 1 and λ = 12 , then we get        d  (d − c)2  1 1− q1 1 1 c+d   + ξ(d) − ξ(x)dx ≤  ξ(c) + 2ξ  4 2 d−c c 16 6  q  q1       c + d  1   − |ξ  (c)|q R1 1, 1, 1 |ξ (c)|q + ξ ×  6 2 2   q1         c + d q  1  1 q q  − |ξ (d)| R1 1, 1, |ξ (d)| + ξ + .  6 2 2 We present the following theorem taken from [157].

Theorem 4.5.2. Let K ⊆ R be an open m-invex subset with respect to η : K × K × (0, 1] → R\{0} for some fixed m ∈ (0, 1], and let c, d ∈ K, c < d with η(d, c, m) > 0. Assume that ξ : K → R is a twice differentiable function   on K such that ξ is integrable on [mc, mc + η(d, c, m)]. If |ξ |q for q > 1 is a generalized (m, h)-preunivex function with respect to η, φ and b and

122

Integral Inequalities and Generalized Convexity

h : [0, 1] → R\{0}, then the following inequality for k-fractional integrals with x ∈ [c, d], λ ∈ [0, 1], α > 0, k > 0 exists:   α +2  η k (x, c, m)  1   p |Iξ,η (α, k; x, λ, m, c, d)| ≤ A3 (k, α, λ, p)  α ( k + 1)η(d, c, m)    q1  1    q q q × m|ξ (c)| + φ(|ξ (x)| − m|ξ (c)| ) h(δ)b(x, c, h(δ))dδ 0    (−1) αk +2 η αk +2 (x, d, m)   +  ( αk + 1)η(d, c, m)   q1    1    q q q h(δ)b(x, d, h(δ))dδ , × m|ξ (d)| + φ(|ξ (x)| − m|ξ (d)| ) 0

q q−1

where p =

and

  1  α p A3 (k, α, λ, p) = 0 δ p  αk + 1 λ − δ k  dδ  1 ,  p( α  k +1)+1     k+kp( α +1) k  p+1  α k[1−( α k(1+p)  k[( αk +1)λ] k +1)λ] β( α , 1 + p) +  , α α(p+1) = k(1+p) α  × 2 F1 (1 − α , 1; p + 2; 1 − ( k + 1)λ)    k+kp( α +1)  k α  α  k[( k +1)λ] 1 β( ( α +1)λ ; k(1+p) , 1 + p), α α k

λ = 0, 0 1 and p = q−1 , then for s ∈ (0, 1] and m ∈ (0, 1], we have the following inequality:    α +2  η k (x, c, m)  1  |Iξ,η (α, k; x, λ, m, c, d)| ≤ A3 (k, α, λ, p) p  α ( k + 1)η(d, c, m)   q1   1    q q q s s δ b(x, c, δ )dδ × m|ξ (c)| + φ(|ξ (x)| − m|ξ (c)| ) 0    (−1) αk +2 η αk +2 (x, d, m)   +  ( αk + 1)η(d, c, m)    q1   1    q q q s s × m|ξ (d)| + φ(|ξ (x)| − m|ξ (d)| ) δ b(x, d, δ )dδ . 0

124

Integral Inequalities and Generalized Convexity

Corollary 4.5.11. In Corollary 4.5.10, if φ : R → R is defined by φ(c) = c and b(., ., .) ≡ 1, then we have 1

|Iξ,η (α, k; x, λ, m, c, d)| ≤ A3 (k, α, λ, p) p   q1   α +2   q q  η k (x, c, m)  (x)| − m|ξ (c)| ) (|ξ  q  m|ξ (c)| + ×  α ( k + 1)η(d, c, m)  (s + 1)

 q1      q q  (−1) αk +2 η αk +2 (x, d, m)   m|ξ  (d)|q + (|ξ (x)| − m|ξ (d)| ) . +  ( αk + 1)η(d, c, m)  (s + 1)

Corollary 4.5.12. In Corollary 4.5.11, if the mapping η(d, c, m) = d − mc together with m = 1, choosing x = c+d 2 , for s ∈ (0, 1], we have the following inequality:      2 αk −1 c+d    (d − c) αk −1 Iξ α, k; 2 , λ, 1, c, d    q1   c+d q q (|ξ ( )| − |ξ (c)| ) 1 (d − c)2 2 ≤ α A3 (k, α, λ, p) p  |ξ  (c)|q + 8( k + 1) (s + 1)  q1     c+d q q (|ξ ( )| − |ξ (d)| ) 2 . + |ξ  (d)|q + (s + 1) Remark 4.5.5. In Corollary 4.5.12,

(1) if λ = 0, then we have     α   k −1 Γ (α + k) k α α ξ c + d − 2  [ J ξ(c) + J ξ(d)] α k ( c+d )− k ( c+d )+   2 2 2 (d − c) k   q1   p1   (|ξ ( c+d )|q − |ξ (c)|q ) (d − c)2 1  q 2  |ξ (c)| + ≤ α 8( k + 1) p( αk + 1) + 1 s+1   q1    q q (|ξ ( c+d  q 2 )| − |ξ (d)| ) , + |ξ (d)| + s+1 (2) if we choose k = α = 1 and λ = 0, then we get      d   c+d 1   − ξ(x)dx ξ   2 d−c c  q1    p1    c+d q q (|ξ ( )| − |ξ (c)| ) 1 (d − c)2 2 |ξ  (c)|q + ≤ 16 2p + 1 s+1  q1     q q (|ξ ( c+d  q 2 )| − |ξ (d)| ) + |ξ (d)| + , s+1

Integral Inequalities for Generalized Preinvex Functions

125

(3) if λ = 1, then we have    ξ(c) + ξ(d) 2 αk −1 Γk (α + k)  α α   − [ J ξ(c) + J ξ(d)] c+d c+d α k ( k ( − +   ) ) 2 2 2 (d − c) k   q1   c+d q q (|ξ ( )| − |ξ (c)| ) 1 (d − c)2 2 ≤ α A3 (k, α, 1, p) p  |ξ  (c)|q + 8( k + 1) s+1  q1     c+d q q (|ξ ( )| − |ξ (d)| ) 2 , + |ξ  (d)|q + s+1 (4) if we choose k = α = λ = 1, then we get    d  ξ(c) + ξ(d)  1   − ξ(x)dx    2 d−c c   p1  1 (d − c)2 1+2p 2 ; 1 + p, 1 + p β ≤ 16 2   q1   c+d q q (|ξ ( )| − |ξ (c)| ) 2 ×  |ξ  (c)|q + s+1  q1     c+d q q (|ξ ( )| − |ξ (d)| ) 2 , + |ξ  (d)|q + s+1 (5) if λ = 13 , then we have      1  ξ(c) + 4ξ c + d + ξ(d) 6 2

 α  2 k −1 Γk (α + k) α α [k J( c+d )− ξ(c) + k J( c+d )+ ξ(d)] − α 2 2 (d − c) k   q1  p1     2 (|ξ ( c+d )|q − |ξ (c)|q ) (d − c) 1  q 2  |ξ (c)| + ≤ α A3 k, α, , p 8( k + 1) 3 s+1   q1    q q (|ξ ( c+d  q 2 )| − |ξ (d)| ) , + |ξ (d)| + s+1

(6) if we choose k = α = 1 and λ = 13 , then we get        d  1 1 c+d   + ξ(d) − ξ(x)dx  ξ(c) + 4ξ  6 2 d−c c

126

Integral Inequalities and Generalized Convexity   q1   p1    c+d q q (|ξ ( )| − |ξ (c)| ) 1 (d − c)2 2  |ξ  (c)|q + A3 1, 1, , p ≤ 16 3 s+1  1 q    q q (|ξ ( c+d  q 2 )| − |ξ (d)| ) , + |ξ (d)| + s+1

(7) if λ = 12 , then we have      1  ξ(c) + 2ξ c + d + ξ(d) 4 2

 α  2 k −1 Γk (α + k) α α [k J( c+d )− ξ(c) + k J( c+d )+ ξ(d)] − α 2 2 (d − c) k   q1 1     c+d q q p (|ξ ( )| − |ξ (c)| ) (d − c)2 1 2  |ξ  (c)|q + ≤ α A3 k, α, , p 8( k + 1) 2 s+1  q1     c+d q q (|ξ ( )| − |ξ (d)| ) 2 , + |ξ  (d)|q + s+1

(8) if we choose k = α = 1 and λ = 12 , then we get        d  1 c+d 1   + ξ(d) − ξ(x)dx  ξ(c) + 2ξ  4 2 d−c c   q1   c+d q q (|ξ ( )| − |ξ (c)| ) (d − c)2 p1 2 β (1 + p, 1 + p)  |ξ  (c)|q + ≤ 16 s+1  q1     c+d q q (|ξ ( )| − |ξ (d)| ) 2 . + |ξ  (d)|q + s+1 Corollary 4.5.13. In Theorem 4.5.2, if we use the generalized (m, tgs) q preunivexity of |ξ |q along with q > 1 and p = q−1 , then for m ∈ (0, 1], we have the following inequality:    α +2  η k (x, c, m)    |Iξ,η (α, k; x, λ, m, c, d)| ≤ A3 (k, α, λ, p)  α ( + 1)η(d, c, m)  1 p



k



q



q



q

× m|ξ (c)| + φ(|ξ (x)| − m|ξ (c)| )



1

0

δ(1 − δ)b(x, c, δ(1 − δ))dδ

 q1

Integral Inequalities for Generalized Preinvex Functions 127   α α  (−1) k +2 η k +2 (x, d, m)   +  ( αk + 1)η(d, c, m)    q1   1    q q q × m|ξ (d)| + φ(|ξ (x)| − m|ξ (d)| ) δ(1 − δ)b(x, d, δ(1 − δ))dδ . 0

Corollary 4.5.14. In Corollary 4.5.13, if φ : R → R is defined by φ(c) = c and b(., ., .) ≡ 1, then we have 1

|Iξ,η (α, k; x, λ, m, c, d)| ≤ A3 (k, α, λ, p) p   q1   α +2   q q  η k (x, c, m)  (x)| − m|ξ (c)| ) (|ξ  m|ξ  (c)|q + ×  α ( k + 1)η(d, c, m)  6

 q1      q q  (−1) αk +2 η αk +2 (x, d, m)  (|ξ (x)| − m|ξ (d)| )  m|ξ  (d)|q + . +  ( αk + 1)η(d, c, m)  6

Corollary 4.5.15. In Corollary 4.5.14, if the mapping η(d, c, m) = d − mc together with m = 1, choosing x = c+d 2 , we have the following inequality:     2 αk −1  c+d    (d − c) αk −1 Iξ α, k; 2 , λ, 1, c, d    q1   2 (|ξ ( c+d )|q − |ξ (c)|q ) 1 (d − c)  q 2 ≤ α A3 (k, α, λ, p) p  |ξ (c)| + 8( k + 1) 6   q1    q q (|ξ ( c+d  q 2 )| − |ξ (d)| ) . + |ξ (d)| + 6

Remark 4.5.6. In Corollary 4.5.15,

(1) if λ = 0, then we have     α   k −1 Γ (α + k) k α α ξ c + d − 2 [k J( c+d )− ξ(c) + k J( c+d )+ ξ(d)] α  k 2 2 2 (d − c)   q1   p1    q q 2 (|ξ ( c+d )| − |ξ (c)| ) (d − c) 1  q 2  |ξ (c)| + ≤ α 8( k + 1) p( αk + 1) + 1 6  q1     c+d q q (|ξ ( )| − |ξ (d)| ) 2 , + |ξ  (d)|q + 6

128

Integral Inequalities and Generalized Convexity

(2) if we choose k = α = 1 and λ = 0, then we get      d   1 c+d   − ξ(x)dx ξ   2 d−c c   q1   p1   (|ξ ( c+d )|q − |ξ (c)|q ) 1 (d − c)2  q 2  |ξ (c)| + ≤ 16 2p + 1 6   q1    q q (|ξ ( c+d  q 2 )| − |ξ (d)| ) , + |ξ (d)| + 6 (3) if λ = 1, then we have    ξ(c) + ξ(d) 2 αk −1 Γk (α + k)  α α  − [k J( c+d )− ξ(c) + k J( c+d )+ ξ(d)] α  k 2 2 2 (d − c)   q1   q q 2 (|ξ ( c+d )| − |ξ (c)| ) 1 (d − c)  q 2 ≤ α A3 (k, α, 1, p) p  |ξ (c)| + 8( k + 1) 6  q1     c+d q q (|ξ ( )| − |ξ (d)| ) 2 , + |ξ  (d)|q + 6 (4) if we choose k = α = λ = 1, then we get    p1   d  (d − c)2   ξ(c) + ξ(d) 1 1   1+2p − 2 ; 1 + p, 1 + p ξ(x)dx ≤ β    2 d−c c 16 2   q1   (|ξ ( c+d )|q − |ξ (c)|q )  q 2 ×  |ξ (c)| + 6  q1     c+d q q (|ξ ( )| − |ξ (d)| ) 2 , + |ξ  (d)|q + 6 (5) if λ = 13 , then we have      1  ξ(c) + 4ξ c + d + ξ(d) 6 2 α

−

2 k −1 Γk (α + k) [k J(αc+d )− ξ(c) + α 2 (d − c) k

  α  J ξ(d)] k ( c+d )+  2

Integral Inequalities for Generalized Preinvex Functions 129   q1   p1   c+d q q (|ξ ( )| − |ξ (c)| ) 1 (d − c)2 2  |ξ  (c)|q + A3 k, α, , p ≤ α 8( k + 1) 3 6  1 q    q q (|ξ ( c+d  q 2 )| − |ξ (d)| ) , + |ξ (d)| + 6 (6) if we choose k = α = 1 and λ = 13 , then we get        d  1 1 c+d   + ξ(d) − ξ(x)dx  ξ(c) + 4ξ  6 2 d−c c   q1  p1    c+d q q (|ξ ( )| − |ξ (c)| ) 1 (d − c)2 2  |ξ  (c)|q + A3 1, 1, , p ≤ 16 3 6   q1    q q (|ξ ( c+d  q 2 )| − |ξ (d)| ) , + |ξ (d)| + 6 (7) if λ = 12 , then we have      1  ξ(c) + 2ξ c + d + ξ(d) 4 2

 α  2 k −1 Γk (α + k) α α − [k J( c+d )− ξ(c) + k J( c+d )+ ξ(d)] α 2 2 (d − c) k   q1 1     c+d q q p (|ξ ( )| − |ξ (c)| ) (d − c)2 1 2  |ξ  (c)|q + ≤ α A3 k, α, , p 8( k + 1) 2 6  q1     c+d q q (|ξ ( )| − |ξ (d)| ) 2 , + |ξ  (d)|q + 6

(8) if we choose k = α = 1 and λ = 12 , then we get        d  1 c+d 1   + ξ(d) − ξ(x)dx  ξ(c) + 2ξ  4 2 d−c c   q1   c+d q q (|ξ ( )| − |ξ (c)| ) (d − c)2 p1 2 β (1 + p, 1 + p)  |ξ  (c)|q + ≤ 16 6  q1     c+d q q (|ξ ( )| − |ξ (d)| ) 2 . + |ξ  (d)|q + 6

130

Integral Inequalities and Generalized Convexity

Corollary 4.5.16. In Theorem 4.5.2, if we use the generalized m − M T  q preunivexity of |ξ |q along with q > 1 and p = q−1 , then for m ∈ (0, 1], we have the following inequality:   α +2  η k (x, c, m)  1   (m|ξ  (c)|q p |Iξ,η (α, k; x, λ, m, c, d)| ≤ A3 (k, α, λ, p)  α ( k + 1)η(d, c, m)     q1 √ √  1   δ δ  + φ(|ξ (x)|q − m|ξ (c)|q ) b x, c,  dδ 2 (1 − δ) 0 2 (1 − δ)    (−1) αk +2 η αk +2 (x, d, m)   (m|ξ  (d)|q  + ( αk + 1)η(d, c, m)     q1  √ √  1   δ δ  b x, d,  dδ  . + φ(|ξ (x)|q − m|ξ (d)|q ) 2 (1 − δ) 0 2 (1 − δ) Corollary 4.5.17. In Corollary 4.5.16, if φ : R → R is defined by φ(c) = c and b(., ., .) ≡ 1, then we have 1

|Iξ,η (α, k; x, λ, m, c, d)| ≤ A3 (k, α, λ, p) p   α +2 1  η k (x, c, m)    m|ξ  (c)|q + π (|ξ  (x)|q − m|ξ  (c)|q ) q ×  α ( + 1)η(d, c, m)  4   k α +2 α +2  q1   (−1) k η k (x, d, m)   π   q q q   . m|ξ (d)| + (|ξ (x)| − m|ξ (d)| ) + ( α + 1)η(d, c, m)  4 k

Corollary 4.5.18. In Corollary 4.5.17, if the mapping η(d, c, m) = d − mc together with m = 1, choosing x = c+d 2 , we have the following inequality:    α −1  2k  c+d    (d − c) αk −1 Iξ α, k; 2 , λ, 1, c, d     q  q1  1 π   c + d  (d − c)2  q q p A3 (k, α, λ, p) ξ |ξ (c)| + ≤ α  − |ξ (c)| 8( k + 1) 4  2   q   q1      π c + d ξ  − |ξ (d)|q + |ξ  (d)|q + .  4  2 Remark 4.5.7. In Corollary 4.5.18,

(1) if λ = 0, then we have     α   k −1 Γ (α + k) k α α ξ c + d − 2  [ J ξ(c) + J ξ(d)] α k ( c+d )− k ( c+d )+   2 2 2 (d − c) k    p1    q  q1  (d − c)2 1 π   c + d   q q ≤ α ξ |ξ (c)| +  − |ξ (c)| 8( k + 1) p( αk + 1) + 1 4  2  q1    q   π   c + d   q q ξ + |ξ (d)| + ,  − |ξ (d)| 4  2

Integral Inequalities for Generalized Preinvex Functions

131

(2) if we choose k = α = 1 and λ = 0, then we get      d   1 c+d   − ξ(x)dx ξ   2 d−c c   q1   p1    q  π   c + d  (d − c)2 1 q |ξ  (c)|q + ξ − |ξ (c)| ≤  16 2p + 1 4  2   q   q1   π   c + d   q q ξ + |ξ (d)| + ,  − |ξ (d)| 4  2

(3) if λ = 1, then we have    ξ(c) + ξ(d) 2 αk −1 Γk (α + k)  α α  − [k J( c+d )− ξ(c) + k J( c+d )+ ξ(d)] α  k 2 2 2 (d − c)    q  q1 2  1 π   c + d  (d − c)  q q p A3 (k, α, 1, p) ξ |ξ (c)| + ≤ α  − |ξ (c)| 8( k + 1) 4  2  q1    q   π   c + d   q q , ξ + |ξ (d)| +  − |ξ (d)| 4  2

(4) if we choose k = α = λ = 1, then we get    p1   d  (d − c)2   ξ(c) + ξ(d) 1 1   1+2p − 2 ; 1 + p, 1 + p ξ(x)dx ≤ β    2 d−c c 16 2    q  q1  π   c + d  q ξ × |ξ  (c)|q + − |ξ (c)|  4  2   q   q1   π   c + d   q q ξ + |ξ (d)| + ,  − |ξ (d)| 4  2

(5) if λ = 13 , then we have      1  ξ(c) + 4ξ c + d + ξ(d) 6 2

 α  2 k −1 Γk (α + k) α α  [ J ξ(c) + J ξ(d)] α k ( c+d )− k ( c+d )+  2 2 (d − c) k   p1    q   q1  (d − c)2 π   c + d  1  q q ≤ α A3 k, α, , p ξ |ξ (c)| +  − |ξ (c)| 8( k + 1) 3 4  2  q1    q   π   c + d   q q ξ + |ξ (d)| + ,  − |ξ (d)| 4  2 −

132

Integral Inequalities and Generalized Convexity

(6) if we choose k = α = 1 and λ = 13 , then we get        d  1 c+d 1   + ξ(d) − ξ(x)dx  ξ(c) + 4ξ  6 2 d−c c 1    q    q1 p  π   c + d  (d − c)2 1  q q A3 1, 1, , p ξ |ξ (c)| + ≤  − |ξ (c)| 16 3 4  2  q1    q   π   c + d   q q , ξ + |ξ (d)| +  − |ξ (d)| 4  2 (7) if λ = 12 , then we have

     1  ξ(c) + 2ξ c + d + ξ(d) 4 2

 α  2 k −1 Γk (α + k) α α − [k J( c+d )− ξ(c) + k J( c+d )+ ξ(d)] α k 2 2 (d − c)   q1  p1    q 2  1 π   c + d  (d − c)  q q |ξ (c)| + A3 k, α, , p ξ ≤ α  − |ξ (c)| 8( k + 1) 2 4  2   q   q1      π c + d ξ  − |ξ (d)|q + |ξ  (d)|q + ,  4  2

(8) if we choose k = α = 1 and λ = 12 , then we get

       d  1 c+d 1   + ξ(d) − ξ(x)dx  ξ(c) + 2ξ  4 2 d−c c    q  q1  π   c + d  (d − c)2 p1 q β (1 + p, 1 + p) |ξ  (c)|q + ξ − |ξ (c)| ≤  16 4  2  q1    q   π   c + d   q q . ξ + |ξ (d)| +  − |ξ (d)| 4  2

Chapter 5 Some Majorization Integral Inequalities for Functions Defined on Rectangles via Strong Convexity

5.1

Introduction

Hardy et al. [58] introduced the notion of majorization and showed that a necessary and sufficient condition that u ≺ v is that there exists a doubly stochastic matrix P˜ such that u = v P˜ . Schur [150] proved that the eigen values majorize the diagonal elements of a positive semidefinite Hermitian matrix. Majorization theory has interesting applications in different fields of mathematics, such as linear algebra, geometry, probability, statistics, group theory, optimization, etc. Niezgoda and Peˇcari´c [115] extended the Hardy– Littlewood–Polya theorem on majorization from convex functions to invex ones and considered some variants for pseudo invex and quasi invex functions. For more details, we can refer to [41, 77, 79–82, 95, 103, 133, 150, 179, 180]. Khan et al. [84] introduced a new class of functions known as coordinate strongly convex function. It is well known that a twice differentiable function ξ : ∆ = [a, b] × [c, d] → R is coordinate strongly convex with respect to µ1 , µ2 > 0 on ∆ if and only if the partial mappings ξy : [a, b] → R defined by ξy (u) = ξ(u, y) and ξx : [c, d] → R defined by ξx (v) = ξ(x, v), satisfied ξy (u) ≥ 2µ1 and ξx (v) ≥ 2µ2 for all u ∈ [a, b] and v ∈ [c, d] (see, [84]). Further, Wu et al. [174] established some defined versions of majorization inequality involving twice differentiable convex functions by using Taylor’s theorem with mean value form of the remainder and also given some interesting applications. In particular, many important inequalities can be found in the literature [1, 30, 31, 74–76, 78, 83]. Recently, Zaheer Ullah et al. [179] established a monotonicity property for the function involving the strongly convex function and proved the classical majorization theorem by using strongly convex functions for majorized n-tuples. Zaheer Ullah et al. [180] obtained integral majorization type and generalized Favard’s inequalities for the class of strongly convex functions. Further, Wu et al. [173] established some majorization integral inequalities and Favard type inequalities for functions defined on rectangles.

DOI: 10.1201/9781003408284-5

133

134

Integral Inequalities and Generalized Convexity

The organization of this chapter is as follows: In Section 5.2, we recall some basic results that are necessary for our main results. In Section 5.3, we extend several integral majorization type and generalized Favard’s inequalities from functions defined on intervals to functions defined on rectangles via strong convexity. The results obtained in this paper are the generalizations of the results given in [173, 180].

5.2

Preliminaries

In this section, we recall some basic results which are useful for our main results. Maligranda et al. [102] proved the following lemma of weighted version of majorization. Lemma 5.2.1. Let ψ be a weight function on [a, b]. (a) If l is decreasing function on [a, b], then 

b

l(u)ψ(u)du a



x

ψ(u)du ≤

a



x

l(u)ψ(u)du a



b a

ψ(u)du for all x ∈ [a, b].

(b) If l is increasing function on [a, b], then 

x

l(u)ψ(u)du a



b a

ψ(u)du ≤



b

l(u)ψ(u)du a



x a

ψ(u)du for all x ∈ [a, b].

Zaheer Ullah et al. [180] obtained the following results for strongly convex functions. Lemma 5.2.2. Let g be a real-valued function defined on [a, b]. Then the following statements are true. (a) If g be a strongly concave function with modulus µ, then (i) the function P1 (u) = g(u)/(u − a) − µu is decreasing on (a, b]; (ii) the function Q1 (u) = g(u)/(b − u) + µu is increasing on [a, b). (b) If g be a strongly convex function with modulus µ, then (i) the function P1 (u) = g(u)/(u−a)−µu is increasing on (a, b], if g(a) = 0; (ii) the function Q1 (u) = g(u)/(b−u)+µu is decreasing on [a, b), if g(b) = 0.

Some Majorization Integral Inequalities for Functions Defined on.....

135

Theorem 5.2.1. Let µ > 0, ξ : [0, ∞) → R be a continuous strongly convex function with modulus µ, and let f, g and w be three positive and integrable functions defined on [a, b] such that  x  x g(u)w(u)du ≤ f (u)w(u)du for all x ∈ [a, b) (5.1) a

and

a



b

g(u)w(u)du = a

Then the following statements are true.



b

f (u)w(u)du.

(5.2)

a

(a) If g is decreasing on [a, b], then we have 

b

ξ{f (u)}w(u)du ≥

a



b

ξ{g(u)}w(u)du + µ

a



b

{g(u) − f (u)}2 w(u)du.

a

(5.3)

(b) If f is increasing on [a, b], then we have 

b a

ξ{g(u)}w(u)du ≥



b

ξ{f (u)}w(u)du + µ

a



a

b

{g(u) − f (u)}2 w(u)du. (5.4)

The following interesting inequality is given by Favard [49]. Theorem 5.2.2. Let f be a nonnegative continuous concave function on [a, b], not identically zero, and let ξ be a convex function on [0, 2f¯], where f¯ = Then, 1 b−a

5.3



2 b−a



b

f (u)du. a

b a

ξ{f (u)}du ≤



1

ξ(sf¯).

0

Majorization Integral Inequalities for Strong Convexity

In this section, first, we prove some majorization integral inequalities for functions defined on rectangles via strong convexity [154].

136

Integral Inequalities and Generalized Convexity

Theorem 5.3.1. Let w and ρ be positive continuous functions on [a, b] and [c, d], respectively, and let f, g and h, k be positive differentiable functions on [a, b] and [c, d], respectively. Suppose that ξ : [0, ∞) × [0, ∞) → R is a strongly convex function with modulus µ > 0 and that  x  x g(u)w(u)du ≤ f (u)w(u)du for all x ∈ [a, b), a



a

y c

k(v)ρ(v)dv ≤ 

b



c

y

h(v)ρ(v)dv for all y ∈ [c, d),

g(u)w(u)du =

a

and



d

k(v)ρ(v)dv = c

 

b

f (u)w(u)du a d

h(v)ρ(v)dv.

c

(a) If g and k are decreasing functions on [a, b] and [c, d], respectively, then 

b a



d

ξ(g(u), k(v))w(u)ρ(v)dudv ≤

c

−µ



a

b



d



b

a



d

ξ(f (u), h(v))w(u)ρ(v)dudv c

(g(u), k(v)) − (f (u), h(v))2 w(u)ρ(v)dudv.

c

(b) If f and h are increasing functions on [a, b] and [c, d], respectively, then 

b

a



d c

ξ(f (u), h(v))w(u)ρ(v)dudv ≤

−µ



a

b

d c



a

b



d

ξ(g(u), k(v))w(u)ρ(v)dudv c

(g(u), k(v)) − (f (u), h(v))2 w(u)ρ(v)dudv.

Proof (a) By the definition of strong convexity, we have ξ(x, y) − ξ(w, z) ≥ + ξ(w, z), (x − w, y − z) + µ(w, z) − (x, y)2 for all (x, y), (w, z) ∈ [0, ∞) × [0, ∞), that is, ∂ξ+ (w, z) ∂ξ+ (w, z) (x − w) + (y − z) ∂w ∂z + µ(w, z) − (x, y)2 , for all (x, y), (w, z) ∈ [0, ∞) × [0, ∞). (5.5)

ξ(x, y) − ξ(w, z) ≥

Some Majorization Integral Inequalities for Functions Defined on.....

137

Putting x = f (u), y = h(v), w = g(u) and z = k(v) in (5.5), we get ξ(f (u), h(v)) − ξ(g(u), k(v))

∂ξ+ (g(u), k(v)) ∂ξ+ (g(u), k(v)) (f (u) − g(u)) + (h(v) − k(v)) ∂g(u) ∂k(v)

≥

+ µ((g(u), k(v)) − (f (u), h(v))2 . 

∂ξ+ (α,β)  , Θ2v (u) ∂α α=g(u),β=k(v)  2 ∂ ξ+ (α,β)  , Θ4v (u) = ∂2α α=g(u),β=k(v)

Suppose Θ1v (u) = Θ3v (u) = Then, we get



∂ξ+ (α,β)  , ∂β α=g(u),β=k(v)  ∂ 2 ξ+ (α,β)  . ∂2β α=g(u),β=k(v)

=

ξ(f (u), h(v)) − ξ(g(u), k(v))

≥ Θ1v (u)(f (u) − g(u)) + Θ2v (u)(h(v) − k(v)) + µ((g(u), k(v)) − (f (u), h(v))2 .

(5.6)

Assume that, F (x) = and G(y) =



x

(f (u) − g(u))w(u)du for all x ∈ [a, b]

a



y c

(h(v) − k(v))ρ(v)dv for all y ∈ [c, d].

From the assumptions of Theorem 5.3.1, we have F (x) ≥ 0, G(y) ≥ 0 for all x ∈ [a, b], y ∈ [c, d] and F (a) = F (b) = G(c) = G(d) = 0. Multiplying both sides by w(u)ρ(v) in (5.6), we have [ξ(f (u), h(v)) − ξ(g(u), k(v))]w(u)ρ(v)

≥ Θ1v (u)[(f (u) − g(u))]w(u)ρ(v) + Θ2v (u)[(h(v) − k(v))]w(u)ρ(v) + µ(g(u), k(v)) − (f (u), h(v))2 w(u)ρ(v).

Integrating both sides above inequality, we have 

a

≥

b





d

[ξ(f (u), h(v)) − ξ(g(u), k(v))]w(u)ρ(v)dudv

c b

a

+





a

+µ

d

Θ1v (u)[(f (u) − g(u))]w(u)ρ(v)dudv

c b d



a

c

b



Θ2v (u)[(h(v) − k(v))]w(u)ρ(v)dudv

c

d

(g(u), k(v)) − (f (u), h(v))2 w(u)ρ(v)dudv.

(5.7)

138

Integral Inequalities and Generalized Convexity Using Fubini’s theorem in above inequality, we get 

b a





≥

d

[ξ(f (u), h(v)) − ξ(g(u), k(v))]w(u)ρ(v)dudv  

c d

b

ρ(v)

c

a



+



b

w(u) a



+µ

b

a



Θ1v (u)dF (u) dv d c

Θ2v (u)dG(v)



du

d

(g(u), k(v)) − (f (u), h(v))2 w(u)ρ(v)dudv.

c

(5.8)

This implies, 

b a





≥

d

[ξ(f (u), h(v)) − ξ(g(u), k(v))]w(u)ρ(v)dudv    b b 1 3   ρ(v) Θv (u)F (u) − Θv (u)g (u)F (u)du dv

c d

a

c



+

b

a

d w(u) Θ2v (u)G(v)c −

a



+µ



b

a





d

c



Θ4v (u)k  (v)G(v)dv du

d

c

(g(u), k(v)) − (f (u), h(v))2 w(u)ρ(v)dudv,

(5.9)

which yields, 

b a



≥− −

d

[ξ(f (u), h(v)) − ξ(g(u), k(v))]w(u)ρ(v)dudv

c





d

c

a

+µ



b

a b d



a

c

b



Θ3v (u)g  (u)F (u)ρ(v)dudv Θ4v (u)k  (v)G(v)w(u)dvdu d

c

(g(u), k(v)) − (f (u), h(v))2 w(u)ρ(v)dudv.

(5.10)

From Lemma 1.1.2, ξ is coordinate strongly convex function, therefore Θ3v (u) ≥ 0, Θ4v (u) ≥ 0. Since, g and k are decreasing functions, therefore g  (u) ≤ 0 and k  (v) ≤ 0. Thus, it follows that −



d c



b a

Θ3v (u)g  (u)F (u)ρ(v)dudv ≥ 0

(5.11)

Some Majorization Integral Inequalities for Functions Defined on..... and −



a

b



d c

Θ4v (u)k  (v)G(v)w(u)dvdu ≥ 0.

139

(5.12)

From (5.10), (5.11) and (5.12), we get 

b

a



−µ

d

[ξ(f (u), h(v)) − ξ(g(u), k(v))]w(u)ρ(v)dudv

c



b

a



d c

(g(u), k(v)) − (f (u), h(v))2 w(u)ρ(v)dudv ≥ 0.

This implies,  b d  ξ(g(u), k(v))w(u)ρ(v)dudv ≤ a

c

−µ



a

b



b

a

d

c



(5.13)

d

ξ(f (u), h(v))w(u)ρ(v)dudv c

(g(u), k(v)) − (f (u), h(v))2 w(u)ρ(v)dudv.

(b) The proof of Theorem 5.3.1(b) is similar to Theorem 5.3.1(a). Remark 5.3.1. If µ = 0, then above theorem reduces to the Theorem 3.1 of [173], i.e., (a) If g and k are decreasing functions on [a, b] and [c, d], respectively, then 

b a



d

ξ(g(u), k(v))w(u)ρ(v)dudv ≤

c



b

a



d

ξ(f (u), h(v))w(u)ρ(v)dudv. c

(b) If f and h are increasing functions on [a, b] and [c, d], respectively, then 

a

b



d c

ξ(f (u), h(v))w(u)ρ(v)dudv ≤



a

b



d

ξ(g(u), k(v))w(u)ρ(v)dudv. c

Theorem 5.3.2. Let w and ρ be positive continuous functions on [a, b] and [c, d], respectively, and let f, g and h, k be positive differentiable functions on [a, b] and [c, d], respectively. Suppose that ξ : [0, ∞) × [0, ∞) → R is a strongly convex function with modulus µ. (a) Let fg and hk be decreasing functions on [a, b] and [c, d], respectively. If f and h are increasing functions on [a, b] and [c, d], respectively, then   b d  f (u) h(v) ξ b , d w(u)ρ(v)dudv a c f (u)w(u)du c h(v)ρ(v)dv a   b d  g(u) k(v) ξ b ≤ , d w(u)ρ(v)dudv a c g(u)w(u)du c k(v)ρ(v)dv a

140

Integral Inequalities and Generalized Convexity    b d  g(u) k(v)  −µ , d  b a c  g(u)w(u)du c k(v)ρ(v)dv a 2   h(v) f (u)  , d − b  f (u)w(u)du h(v)ρ(v)dv  a

c

× w(u)ρ(v)dudv.

(5.14)

(b) Let fg and hk be increasing functions on [a, b] and [c, d], respectively. If g and k are increasing functions on [a, b] and [c, d], respectively, then   b d  g(u) k(v) ξ b , d w(u)ρ(v)dudv a c g(u)w(u)du k(v)ρ(v)dv a c   b d  h(v) f (u) ξ b ≤ , d w(u)ρ(v)dudv a c f (u)w(u)du c h(v)ρ(v)dv a    b d  g(u) k(v)  −µ , d  b a c  g(u)w(u)du k(v)ρ(v)dv a c 2   h(v) f (u)  , d − b  f (u)w(u)du c h(v)ρ(v)dv  a × w(u)ρ(v)dudv.

(5.15)

Proof (a) Applying Lemma 5.2.1(a) with substitution ψ(u) = g(u)w(u) and l(u) = f (u) g(u) , we have 

b

f (u)w(u)du a

≤



which yields,  

x

≤



a

g(u)w(u)du a

a

b

x

x

f (u)w(u)du

x

a



a 

g(u) g(u)w(u)du b a





b a

g(u)w(u)du for all x ∈ [a, b],

w(u)du

f (u) f (u)w(u)du



w(u)du for all x ∈ [a, b].

(5.16)

Some Majorization Integral Inequalities for Functions Defined on..... Also, substituting ψ(v) = k(v)ρ(v) and l(v) = we have  d  y h(v)ρ(v)dv k(v)ρ(v)dv c

≤



c

y

h(v)ρ(v)dv

c

which yields,   ≤



c

d

y

c



k(v) k(v)ρ(v)dv d c



in Lemma 5.2.1(a),

d

c

y

c



h(v) k(v)

141

k(v)ρ(v)dv for all y ∈ [c, d],

ρ(v)dv

h(v) h(v)ρ(v)dv



ρ(v)dv for all y ∈ [c, d].

(5.17)

Additionally, it is easy to observe that from (5.16) and (5.17), we have    b  b g(u) f (u) w(u)du = w(u)du, b b a a g(u)w(u)du f (u)w(u)du a a (5.18) 

d c



d c

k(v) k(v)ρ(v)dv



ρ(v)dv =



c

d



d c

h(v) h(v)ρ(v)dv



ρ(v)dv. (5.19)

Applying (5.16), (5.17), (5.18) and (5.19) in Theorem 5.3.1(b), we have   b d  f (u) h(v) ξ b , d w(u)ρ(v)dudv a c f (u)w(u)du h(v)ρ(v)dv a c   b d  g(u) k(v) ξ b , d w(u)ρ(v)dudv ≤ a c g(u)w(u)du c k(v)ρ(v)dv a    b d  k(v) g(u)  −µ , d  b a c  g(u)w(u)du c k(v)ρ(v)dv a 2   h(v) f (u)  , d − b  f (u)w(u)du c h(v)ρ(v)dv  a × w(u)ρ(v)dudv.

(b) The proof of Theorem 5.3.2(b) is similar to Theorem 5.3.2(a).

(5.20)

142

Integral Inequalities and Generalized Convexity

Remark 5.3.2. If µ = 0, then above theorem reduces to the Theorem 3.2 of [173], i.e., (a) Let fg and hk be decreasing functions on [a, b] and [c, d], respectively. If f and h are increasing functions on [a, b] and [c, d], respectively, then   b d  f (u) h(v) ξ b , d w(u)ρ(v)dudv a c f (u)w(u)du c h(v)ρ(v)dv a   b d  g(u) k(v) ξ b ≤ , d w(u)ρ(v)dudv. a c g(u)w(u)du k(v)ρ(v)dv a c (b) Let fg and hk be increasing functions on [a, b] and [c, d], respectively. If g and k are increasing functions on [a, b] and [c, d], respectively, then   b d  g(u) k(v) ξ b , d w(u)ρ(v)dudv a c g(u)w(u)du k(v)ρ(v)dv a c   b d  f (u) h(v) ξ b , d w(u)ρ(v)dudv. ≤ a c f (u)w(u)du c h(v)ρ(v)dv a

Next, we establish some Favard’s type inequalities for functions defined on rectangles via strong convexity [154]. Theorem 5.3.3. (a) Let g and k be strongly concave functions with modulus µ1 and µ2 on [a, b] and [c, d], respectively, such that f (u) = g(u) − µ1 u(u − a) and h(v) = k(v) − µ2 v(v − c) are positive increasing functions. Also suppose ξ be a strongly convex function with modulus µ on [0, 2g¯1 ] × [0, 2k¯1 ], z¯1 = a(1− ζ) + bζ, z¯2 = c(1 − τ ) + dτ, g¯1 = (b−a) ab f (u)w(u)du (d−c) cd h(v)ρ(v)dv b d and k¯1 = . 2

a

(u−a)w(u)du

2

Then

c

(v−c)ρ(v)dv

 b d 1 ξ(f (u), h(v))w(u)ρ(v)dudv (b − a)(d − c) a c  1 1 ξ(2g¯1 ζ, 2k¯1 τ )w(¯ z1 )ρ(¯ z2 )dζdτ ≤ 0

−µ



0 1

0



1

0

(2g¯1 ζ, 2k¯1 τ ) − (g(¯ z1 ) − µ1 z¯1 (b − a)ζ, k(¯ z2 )

− µ2 z¯2 (d − c)τ )2 w(¯ z1 )ρ(¯ z2 )dζdτ.

(5.21)

If g and k be strongly convex functions with modulus µ1 and µ2 on [a, b] and [c, d], respectively, such that f (u) = g(u) − µ1 u(u − a) and h(v) = k(v) − µ2 v(v − c) are positive increasing functions and g(a) = k(c) = 0, then the reverse inequality in (5.21) holds.

Some Majorization Integral Inequalities for Functions Defined on.....

143

(b) Let g and k be strongly concave functions with modulus µ1 and µ2 on [a, b] and [c, d], respectively, such that f (u) = g(u) + µ1 u(b − u) and h(v) = k(v) + µ2 v(d − v) are positive decreasing functions. Also suppose ¯1 = ξ be a strongly convex function with modulus µ on[0, 2g¯2 ]×[0, 2k¯2 ], ω (b−a) ab f (u)w(u)du ¯ b and k2 = aζ + b(1 − ζ), ω ¯ 2 = cτ + d(1 − τ ), g¯2 = 2

 (d−c) cd h(v)ρ(v)dv d . 2 c (d−v)ρ(v)dv

a

(b−u)w(u)du

Then

 b d 1 ξ(f (u), h(v))w(u)ρ(v)dudv (b − a)(d − c) a c  1 1 ≤ ξ(2g¯2 ζ, 2k¯2 τ )w(¯ ω1 )ρ(¯ ω2 )dζdτ 0

−µ



0 1



0

1

0

(2g¯2 ζ, 2k¯2 τ ) − (g(¯ ω1 ) − µ1 ω ¯ 1 (b − a)ζ, k(¯ ω2 )

− µ2 ω ¯ 2 (d − c)τ )2 w(¯ ω1 )ρ(¯ ω2 )dζdτ.

(5.22)

If g and k be strongly convex functions with modulus µ1 and µ2 on [a, b] and [c, d], respectively, such that f (u) = g(u) + µ1 u(b − u) and h(v) = k(v) + µ2 v(d − v) are positive increasing functions and g(a) = k(c) = 0, then the reverse inequality in (5.22) holds. Proof (a) From Lemma 5.2.2, we know that the function P1 (u) = P2 (v) = and l(u)

k(v) v−c − µ2 v is decreasing g(u) = u−a − µ1 u in Lemma



b

f (u)w(u)du a

≤ that is,  x a



x



a

then substituting ψ(u) = (u − a)w(u) 5.2.1(a), we get

(u − a)w(u)du 

f (u)w(u)du

a



b a

(u − a)w(u)du for all x ∈ [a, b],

x

a

f (u)w(u)du for all x ∈ [a, b].

Also, substitute ψ(v) = (v − c)ρ(v) and l(v) = 5.2.1(a), we obtain  y  d h(v)ρ(v)dv (v − c)ρ(v)dv ≤



c

− µ1 u and

x

(u − a) 2g¯1 w(u)du ≤ (b − a)

c

g(u) u−a

c

y

h(v)ρ(v)dv



c

k(v) v−c

(5.23)

− µ2 v in Lemma

d

(v − c)ρ(v)dv for all y ∈ [c, d],

144

Integral Inequalities and Generalized Convexity that is,  y c

(v − c) ¯ 2k1 ρ(v)dv ≤ (d − c)



c

y

h(v)ρ(v)dv for all y ∈ [c, d].

(5.24)

Since f and h are increasing functions, therefore from (5.23), (5.24) and Theorem 5.3.1(b), we obtain 

b

a



d

ξ(f (u), h(v))w(u)ρ(v)dudv c b



 u−a v−c ¯ 2g¯1 , 2k1 w(u)ρ(v)dudv b−a d−c a c 2   b  d   u−a  v−c ¯   −µ  b − a 2g¯1 , d − c 2k1 − (f (u), h(v)) w(u)ρ(v)dudv.

≤





a

d

ξ

c

(5.25)

Multiplying on both sides by 1 (b − a)(d − c)



b a



1 (b−a)(d−c)

in (5.25), we get

d

ξ(f (u), h(v))w(u)ρ(v)dudv c b



 u−a v−c ¯ 2g¯1 , 2k1 w(u)ρ(v)dudv ξ b−a d−c a c 2   b  d   u−a  v−c ¯ µ   2 g ¯ 2 k , − (f (u), h(v)) − 1 1   (b − a)(d − c) a c b−a d−c

1 ≤ (b − a)(d − c)





d

× w(u)ρ(v)dudv.

Applying change of variable technique in above inequality, we have  b d 1 ξ(f (u), h(v))w(u)ρ(v)dudv (b − a)(d − c) a c      2g¯1  2k¯1 1 (b − a)x (d − c)y ≤ ξ(x, y)w a + ρ c+ dxdy 2g¯1 4g¯1 k¯1 0 2k¯1 0     2  2g¯1  2k¯1    µ (x, y) − f a + (b − a)x , h c + (d − c)y  −   2g¯1 4g¯1 k¯1 0 2k¯1 0     (d − c)y (b − a)x ρ c+ dxdy. (5.26) ×w a+ 2g¯1 2k¯1 Since,       (b − a)x (b − a)x (b − a)x (b − a)x f a+ =g a+ − µ1 a + 2g¯1 2g¯1 2g¯1 2g¯1 (5.27)

Some Majorization Integral Inequalities for Functions Defined on.....

145

and       (d − c)y (d − c)y (d − c)y (d − c)y h c+ . =k c+ − µ2 c + 2k¯1 2k¯1 2k¯1 2k¯1 (5.28) From (5.26), (5.27) and (5.28), we get  b d 1 ξ(f (u), h(v))w(u)ρ(v)dudv (b − a)(d − c) a c      2g¯1  2k¯1 1 (b − a)x (d − c)y ≤ ξ(x, y)w a + ρ c+ dxdy 2g¯1 4g¯1 k¯1 0 2k¯1 0     2g¯1  2k¯1   µ (x, y) − g a + (b − a)x −  2g¯1 4g¯1 k¯1 0 0     (b − a)x (b − a)x (d − c)y − µ1 a + ,k c + 2g¯1 2g¯1 2k¯1   2   (d − c)y (d − c)y  (b − a)x  − µ2 c +  w a + 2g¯1 2k¯1 2k¯1   (d − c)y ×ρ c+ dxdy. (5.29) 2k¯1 Applying change of variable technique, we obtain  b d 1 ξ(f (u), h(v))w(u)ρ(v)dudv (b − a)(d − c) a c  1 1 ξ(2g¯1 ζ, 2k¯1 τ )w(a + (b − a)ζ)ρ(c + (d − c)τ )dζdτ ≤ 0

−µ



0 1

0



1

0

(2g¯1 ζ, 2k¯1 τ ) − (g(a + (b − a)ζ)

− µ1 (a + (b − a)ζ)(b − a)ζ, k(c + (d − c)τ )

− µ2 (c + (d − c)τ )(d − c)τ )2 w(a + (b − a)ζ)ρ(c + (d − c)τ )dζdτ. (5.30)

This implies,  b d 1 ξ(f (u), h(v))w(u)ρ(v)dudv (b − a)(d − c) a c  1 1 ξ(2g¯1 ζ, 2k¯1 τ )w(¯ z1 )ρ(¯ z2 )dζdτ ≤ 0

−µ



0 1

0



1

0

(2g¯1 ζ, 2k¯1 τ ) − (g(¯ z1 ) − µ1 z¯1 (b − a)ζ, k(¯ z2 )

− µ2 z¯2 (d − c)τ )2 w(¯ z1 )ρ(¯ z2 )dζdτ. Similarly, we can proved the reverse inequality of (5.21).

(5.31)

146

Integral Inequalities and Generalized Convexity

(b) The proof of Theorem 5.3.3(b) is similar to Theorem 5.3.3(a). Corollary 5.3.1. If µ = 0, then we have the following new results: (a)  b d 1 ξ(f (u), h(v))w(u)ρ(v)dudv (b − a)(d − c) a c  1 1 ξ(2g¯1 ζ, 2k¯1 τ )w(¯ z1 )ρ(¯ z2 )dζdτ. ≤ 0

0

(b)  b d 1 ξ(f (u), h(v))w(u)ρ(v)dudv (b − a)(d − c) a c  1 1 ξ(2g¯2 ζ, 2k¯2 τ )w(¯ ω1 )ρ(¯ ω2 )dζdτ. ≤ 0

0

Theorem 5.3.4. Let f (u) = g(u) − µ1 u(u − a), h(v) = k(v) − µ2 v(v − c) are h be decreasing functions on (0, 1). Also increasing functions on (0, 1) and Pf , Q suppose f, h, P, Q, w and ρ are positive functions on (0, 1), and f w, hρ, P w and Qρ are integrable on (0, 1) such that 1 1 f (u)w(u)du h(v)ρ(v)dv 0 φ = 1 ≥ 0 and ϕ =  01 ≥ 0. (5.32) P (u)w(u)du Q(v)ρ(v)dv 0 0 Suppose ξ be a strongly convex function with modulus µ. Then,  1 1 ξ(mf (u), nh(v))w(u)ρ(v)dudv 0

0

≤ −µ



1

0  1 0



1

0  1 0

ξ(mφP (u), nϕQ(v))w(u)ρ(v)dudv (mφP (u), nϕQ(v)) − (mf (u), nh(v))2 w(u)ρ(v)dudv

for all m, n > 0. Proof From P > 0 and (5.32), substituting ψ(u) = P (u)w(u) and l(u) = f (u)/P (u), in Lemma 5.2.1(a), we get  x  x mφP (u)w(u)du ≤ mf (u)w(u)du. (5.33) 0

0

Similarly, Q > 0 and (5.32), substituting ψ(v) = Q(v)ρ(v) and l(v) = h(v)/Q(v), in Lemma 5.2.1(a), we get  y  y nϕQ(v)ρ(v)dv ≤ nh(v)ρ(v)dv. (5.34) 0

0

Some Majorization Integral Inequalities for Functions Defined on.....

147

Since f and h are increasing functions, therefore by using Theorem 5.3.1(b), we get  1 1 ξ(mf (u), nh(v))w(u)ρ(v)dudv 0

≤

0 1



0

−µ





1

ξ(mφP (u), nϕQ(v))w(u)ρ(v)dudv 0 1



0

1

(mφP (u), nϕQ(v)) − (mf (u), nh(v))2 w(u)ρ(v)dudv.

0

Corollary 5.3.2. If µ = 0, then we have the following new result:  1 1 ξ(mf (u), nh(v))w(u)ρ(v)dudv 0

≤

0 1



0

−µ





1

ξ(mφP (u), nϕQ(v))w(u)ρ(v)dudv 0 1



0

1

(mφP (u), nϕQ(v)) − (mf (u), nh(v))2 w(u)ρ(v)dudv

0

for all m, n > 0. Theorem 5.3.5. Let ξ : [0, ∞) → R be a continuous strongly convex function with modulus µ, f (u) = g(u) − µ1 u(u − a) and h(v) = k(v) − µ2 v(v − c), P, w and Q, ρ are positive integrable functions on [a,b] and [c,d], respecf (u) P (u) h(v) , z2 (u) =  b P (u)w(u)du , ω1 (v) =  d h(v)ρ(v)dv , tively, z1 (u) =  b f (u)w(u)du a

and ω2 (v) =

d c

Q(v) . Q(v)ρ(v)dv

a

c

Then the following statements are true.

(a) If f and h are increasing on [a, b] and [c, d], respectively, and f /P and h/Q are decreasing on [a, b] and [c, d], respectively, then 

b

a

≤





d

ξ(z1 (u), ω1 (v))w(u)ρ(v)dudv c b

a

−µ



d

c  b a

ξ(z2 (u), ω2 (v))w(u)ρ(v)dudv 

d c

(z2 (u), ω2 (v)) − (z1 (u), ω1 (v))2 w(u)ρ(v)dudv. (5.35)

(b) If P and Q are increasing on [a, b] and [c, d], respectively, and f /P and h/Q are increasing on [a, b] and [c, d], respectively, then 

a

b



d

ξ(z2 (u), ω2 (v))w(u)ρ(v)dudv c

148

Integral Inequalities and Generalized Convexity ≤



a

b



−µ

d

ξ(z1 (u), ω1 (v))w(u)ρ(v)dudv c b

a



d

(z2 (u), ω2 (v)) − (z1 (u), ω1 (v))2 w(u)ρ(v)dudv. (5.36)

c

Proof (a) Since P > 0, then substituting ψ(u) = P (u)w(u) and l(u) = f (u)/P (u), in Lemma 5.2.1(a), we get  x  x z2 (u)w(u)du ≤ z1 (u)w(u)du. (5.37) a

a

Similarly, Q > 0 and (5.32), substituting ψ(v) = Q(v)ρ(v)andl(v) = h(v)/Q(v), in Lemma 5.2.1(a), we get  y  y ω2 (v)ρ(v)dv ≤ ω1 (u)ρ(v)dv. (5.38) c

c

f and h are increasing functions, then applying Theorem 5.3.1(b), we have  b d ξ(z1 (u), ω1 (v))w(u)ρ(v)dudv a

≤



c b

a

−µ





d

ξ(z2 (u), ω2 (v))w(u)ρ(v)dudv c b

a



d c

(z2 (u), ω2 (v)) − (z1 (u), ω1 (v))2 w(u)ρ(v)dudv. (5.39)

(b) The proof of Theorem 5.3.5(b) is similar to Theorem 5.3.5(a). Corollary 5.3.3. If µ = 0, then we have the following new result: (a) 

b a



≤

d

ξ(z1 (u), ω1 (v))w(u)ρ(v)dudv c



b

a



d

ξ(z2 (u), ω2 (v))w(u)ρ(v)dudv. c

(b) 

b

a



≤

d

ξ(z2 (u), ω2 (v))w(u)ρ(v)dudv c



a

b



d

ξ(z1 (u), ω1 (v))w(u)ρ(v)dudv. c

Chapter 6 Hermite–Hadamard Type Inclusions for Interval-Valued Generalized Preinvex Functions

6.1

Introduction

Interval analysis is a new and growing branch of applied mathematics. It is an approach to computing that treats an interval as a new kind of numbers. Moore [109] computed arbitrarily sharp upper and lower bounds on exact solutions of many problems in applied mathematics by using interval arithmetic, interval-valued functions and integrals of interval-valued functions. Moore [109] showed that if a real-valued function ξ(x) satisfies an ordinary Lipschitz condition in K, |ξ(x) − ξ(y)|  L|x − y| for x, y ∈ K, then the united extension of ξ is a Lipschitz interval extension in K. Bhurjee and Panda [18] defined the interval-valued function in the parametric form and developed a methodology to study the existence of the solution of a general interval optimization problem. Lupulescu [100] introduced the differentiability and integrability for the interval-valued functions on time scales by using the concept of the generalized Hukuhara difference. Chalco-Cano et al. [26] proposed a new Ostrowski type inequalities for gH– differentiable interval-valued functions and obtained generalization of the class of real functions which is not necessarily differentiable. Chalco-Cano et al. [26] obtained error bounds to quadrature rules for gH–differentiable intervalvalued functions. Further, Roy and Panda [144] introduced the concept of µ-monotonic property of interval-valued function in the higher dimension and derived some results by using generalized Hukuhara differentiability. For more deatils of interval-valued functions, we refer to [19, 20, 67, 96, 101, 144] and references therein. An et al. [4] introduced (h1 , h2 )-convex interval-valued function and obtained some interval Hermite–Hadamard type inequalities. Further, Budak et al. [25] established the Hermite–Hadamard inequality for the convex intervalvalued function and for the product of two convex interval-valued functions. Zhao et al. [185] introduced the notion of harmonical h-convexity for intervalvalued functions and proved some new Hermite–Hadamard type inequalities for the interval Riemann integral. Recently, Zhao et al. [183, 186] proposed DOI: 10.1201/9781003408284-6

149

150

Integral Inequalities and Generalized Convexity

the notion of interval-valued convex functions on coordinates and established Hermite–Hadamard type inequalities for these interval-valued coordinated convex functions. Further, Budak et al. [24] described a new concept of interval-valued fractional integrals on coordinates and investigated Hermite– Hadamard type inequalities for interval-valued coordinated convex functions using these fractional integrals. Kara et al. [70] proved Hermite–HadamardFej´ er type inclusions for the product of two interval-valued convex functions on coordinates. The organization of this chapter is as follows: In Section 6.2, we recall some basic results that are necessary for our main results. In Section 6.3, we introduce the concept of (h1 , h2 )-preinvex interval-valued function and establish the Hermite–Hadamard inequality for preinvex interval-valued functions and for the product of two preinvex interval-valued functions via intervalvalued Riemann–Liouville fractional integrals. In Section 6.4, we define preinvex interval-valued functions on coordinates in a rectangle from the plane and investigate Hermite–Hadamard type inclusions for coordinated preinvex interval-valued functions. Further, we present Hermite–Hadamard type inclusions for the product of two interval-valued preinvex functions on coordinates. In Section 6.5, we define harmonically h-preinvexity of interval-valued functions and prove fractional Hermite–Hadamard type inclusions for harmonically h-preinvex interval-valued functions. In this way, these findings include several well-known results and newly obtained results of the existing literature as special cases. Some examples are given to confirm our theoretical results.

6.2

Preliminaries

In this section, we mention some definitions and related results required for this chapter. The following theorem gives a relation between interval-Riemann integrable (IR-integrable) and Riemann integrable (R-integrable) functions [110]. Theorem 6.2.1. Let ξ : [c, d] → RI be an interval-valued function such that ξ(δ) = [ξ(δ), ξ(δ)]. ξ ∈ IR([c,d]) if and only if ξ(δ), ξ(δ) ∈ R([c,d]) and (IR)



d



ξ(δ)dδ = (R) c



c

d

ξ(δ)dδ, (R)



d



ξ(δ)dδ ,

c

where R([c,d]) denotes the collection of R-integrable functions. The following interval-valued left-sided and right-sided Riemann–Liouville fractional integral of function are defined by Lupulescu [101] and Budak et al. [25], respectively.

Hermite–Hadamard Type Inclusions for Interval-Valued Generalized...

151

Definition 6.2.1. Let ξ : [c, d] → RI be an interval-valued function such that ξ(δ) = [ξ(δ), ξ(δ)] and ξ ∈ IR([c,d]) . The interval-valued left-sided Riemann– Liouville fractional integral of function ξ is defined by  x 1 Jcα+ ξ(x) = (IR) (x − δ)(α−1) ξ(δ)dδ, x > c, α > 0, Γ(α) c where Γ(α) is the Gamma function. Definition 6.2.2. Let ξ : [c, d] → RI be an interval-valued function such that ξ(δ) = [ξ(δ), ξ(δ)] and ξ ∈ IR([c,d]) . The interval-valued right-sided Riemann– Liouville fractional integral of function ξ is defined by  d 1 α (IR) (δ − x)(α−1) ξ(δ)dδ, x < d, α > 0, Jd− ξ(x) = Γ(α) x where Γ(α) is the Gamma function. The following result is given by Budak et al. [25]. Corollary 6.2.1. If ξ : [c, d] → RI is an interval-valued function such that ξ(δ) = [ξ(δ), ξ(δ)] with ξ(δ), ξ(δ) ∈ R([c,d]) , then we have Jcα+ ξ(x) = [Jcα+ ξ(x), Jcα+ ξ(x)] and Jdα− ξ(x) = [Jdα− ξ(x)Jdα− ξ(x)]. Sadowska [147] gave the following concept of convex interval-valued functions. Definition 6.2.3. Let ξ : [c, d] → R+ I be an interval-valued function such that ξ(x) = [ξ(x), ξ(x)]. We say that ξ is convex interval-valued function if ξ(δx + (1 − δ)y) ⊇ δξ(x) + (1 − δ)ξ(y), ∀ δ ∈ [0, 1] and ∀ x, y ∈ [c, d].

6.3

Hermite–Hadamard Type Inclusions for Interval-Valued Preinvex Functions

In this section, first, we give the definition of interval-valued h-preinvex function and discuss some special cases of interval-valued h-preinvex functions [160]. Definition 6.3.1. Let h : [a, b] → R be a nonnegative function, (0, 1) ⊆ [a, b] and h = 0. Let K ⊆ R be a invex set with respect to η : K × K → R, ξ(x) = [ξ(x), ξ(x)] be an interval-valued function defined on K. We say that ξ is h−preinvex at x with respect to η if ξ(y + δη(x, y)) ⊇ h(δ)ξ(x) + h(1 − δ)ξ(y), ∀ δ ∈ [0, 1] and ∀ x, y ∈ K.

152

Integral Inequalities and Generalized Convexity

Now, we discuss some special cases of interval-valued h−preinvex functions. (1) If h(δ) = 1, then we have the definition of interval-valued P −preinvex functions. (2) If h(δ) = δ, then we have the definition of interval-valued preinvex functions. (3) If h(δ) = δ −1 , then we have the definition of interval-valued Q−preinvex functions. (4) If h(δ) = δ s with s ∈ (0, 1), then we have the definition of interval-valued s−preinvex functions. Example 6.3.1. K = [1, 2], ξ(x) = [x, 10 − ex ], η(x, y) = x − 2y and h(δ) = δ. Then ξ is h−preinvex interval-valued function with respect to η. Theorem 6.3.1. Let h : [a, b] → R be a nonnegative function, (0, 1) ⊆ [c, d] and h = 0. Let K be an invex subset of R with respect to η : K × K → R and ξ be an interval-valued function defined on K. Then ξ is h−preinvex at x if and only if ξ(y + δη(x, y)) ≤ h(δ)ξ(x) + h(1 − δ)ξ(y) and

ξ(y + δη(x, y)) ≥ h(δ)ξ(x) + h(1 − δ)ξ(y), ∀ δ ∈ [0, 1] and ∀ x ∈ K. Now, we establish the Hermite–Hadamard inequalities for the preinvex interval-valued functions [160]. Theorem 6.3.2. Let K ⊆ R be an open invex subset with respect to η : K × K → R and c, d ∈ K with c < c+η(d, c). If ξ : [c, c+η(d, c)] → R+ I is a preinvex interval-valued function such that ξ(δ) = [ξ(δ), ξ(δ)]. ξ ∈ L1 [c, c + η(d, c)] and η satisfies Condition C and α > 0, then we have   Γ(α + 1) α η(d, c) α ⊇ α [J + ξ(c + η(d, c)) + J(c+η(d,c)) ξ c+ − ξ(c)] 2 2η (d, c) c ξ(c) + ξ(c + η(d, c)) ξ(c) + ξ(d) ⊇ ⊇ . (6.1) 2 2 Proof Since ξ is preinvex interval-valued function, we have   ξ(x) + ξ(y) 1 , ∀ x, y ∈ [c, c + η(d, c)]. ξ x + η(y, x) ⊇ 2 2

(6.2)

Using x = c + (1 − δ)η(d, c), y = c + δη(d, c) and Condition C in (6.2), we get 1 ξ(c + (1 − δ)η(d, c) + η(c + δη(d, c), c + (1 − δ)η(d, c))) 2 ξ(c + (1 − δ)η(d, c)) + ξ(c + δη(d, c)) . ⊇ 2

Hermite–Hadamard Type Inclusions for Interval-Valued Generalized... This implies,   1 ξ(c + (1 − δ)η(d, c)) + ξ(c + δη(d, c)) ξ c + η(d, c) ⊇ . 2 2

153

(6.3)

Multiplying by δ α−1 , α > 0 on both sides in (6.3), we have   δ α−1 1 [ξ(c + (1 − δ)η(d, c)) + ξ(c + δη(d, c))]. (6.4) δ α−1 ξ c + η(d, c) ⊇ 2 2 Integrating above inequality on [0, 1], we get   1 δ α−1 ξ c + η(d, c) dδ 2 0    1  1 1 ⊇ (IR) δ α−1 ξ(c + (1 − δ)η(d, c))dδ + (IR) δ α−1 ξ(c + δη(d, c))dδ . 2 0 0 (6.5)

(IR)



1

Applying Theorem 6.2.1 in above relation, we get    1 1 (IR) δ α−1 ξ c + η(d, c) dδ 2 0        1  1 1 1 = (R) δ α−1 ξ c + η(d, c) dδ, (R) δ α−1 ξ c + η(d, c) dδ 2 2 0 0        1  1 1 1 α−1 α−1 = ξ c + η(d, c) (R) δ dδ, ξ c + η(d, c) (R) δ dδ 2 2 0 0      1 1 1 1 ξ c + η(d, c) , ξ c + η(d, c) = α 2 α 2   1 1 = ξ c + η(d, c) . (6.6) α 2 (IR)



1

δ α−1 ξ(c + δη(d, c))dδ

0



= (R)



0

1

δ

α−1

ξ(c + δη(d, c))dδ, (R)



0

1

δ

α−1



ξ(c + δη(d, c))dδ .

154

Integral Inequalities and Generalized Convexity

This implies, (IR) 



1

δ α−1 ξ(c + δη(d, c))dδ

0

1 = (R) α η (d, c) 1 (R) η α (d, c) = =

Γ(α) η α (d, c)



c+η(d,c)

(v − c)α−1 ξ(v)dv,

c



c+η(d,c)

(v − c)

c

α−1

ξ(v)dv



α α [J(c+η(d,c)) − ξ(c), J(c+η(d,c))− ξ(c)]

Γ(α) α J − ξ(c). α η (d, c) (c+η(d,c))

(6.7)

Similarly, (IR)



1

δ α−1 ξ(c + (1 − δ)η(d, c))dδ =

0

Γ(α) α J + ξ(c α η (d, c) c

+ η(d, c)).

Using (6.6), (6.7) and (6.8) in (6.5), we have   Γ(α + 1) α 1 α [J + ξ(c + η(d, c)) + J(c+η(d,c)) ξ c + η(d, c) ⊇ α − ξ(c)]. 2 2η (d, c) c

(6.8)

(6.9)

Now, we prove the second pair of inequality. Since, ξ is an interval-valued preinvex function on [c, c + η(d, c)]. Therefore, ξ(c + δη(d, c)) = ξ(c + η(d, c) + (1 − δ)η(c, c + η(d, c))) ⊇ δξ(c + η(d, c)) + (1 − δ)ξ(c)

(6.10)

ξ(c + (1 − δ)η(d, c)) = ξ(c + η(d, c) + δη(c, c + η(d, c))) ⊇ (1 − δ)ξ(c + η(d, c)) + δξ(c).

(6.11)

and

Adding (6.10) and (6.11), we have ξ(c + δη(d, c)) + ξ(c + (1 − δ)η(d, c)) ⊇ ξ(c) + ξ(c + η(d, c)).

(6.12)

Multiplying by δα−1 and integrating on [0, 1], we have (IR)



1

δ α−1 ξ(c + δη(d, c))dδ + (IR) 0

⊇ (IR)



0



0

1

1

δ α−1 ξ(c + (1 − δ)η(d, c))dδ

δ α−1 [ξ(c) + ξ(c + η(d, c))].

(6.13)

Hermite–Hadamard Type Inclusions for Interval-Valued Generalized...

155

Applying Theorem 6.2.1 in above relation, we get  1 δ α−1 [ξ(c) + ξ(c + η(d, c))] (IR) 0   1 = (R) δ α−1 [ξ(c) + ξ(c + η(d, c))]dδ, 0

(R) 



1

δ α−1 [ξ(c) + ξ(c + η(d, c))]dδ 0

= [ξ(c) + ξ(c + η(d, c))](R) [ξ(c) + ξ(c + η(d, c))](R) 



1



δ α−1 dδ, 0



1

δ 0

α−1

dδ



1 1 = [ξ(c) + ξ(c + η(d, c))], [ξ(c) + ξ(c + η(d, c))] α α 1 = [ξ(c) + ξ(c + η(d, c))]. α

 (6.14)

Using (6.7), (6.8) and (6.14) in (6.13), we have Γ(α + 1) α α [J + ξ(c + η(d, c)) + J(c+η(d,c)) − ξ(c)] 2η α (d, c) c ξ(c) + ξ(d) ξ(c) + ξ(c + η(d, c)) ⊇ . ⊇ 2 2

(6.15)

From (6.9) and (6.15), we get   η(d, c) Γ(α + 1) α α ξ c+ ⊇ α [J + ξ(c + η(d, c)) + J(c+η(d,c)) − ξ(c)] 2 2η (d, c) c ξ(c) + ξ(c + η(d, c)) ξ(c) + ξ(d) ⊇ ⊇ . 2 2 This completes the proof. Corollary 6.3.1. If α = 1, then Theorem 6.3.2 reduces to the following result:    c+η(d,c) 1 η(d, c) ⊇ ξ(δ)dδ ξ c+ 2 η(d, c) c ξ(c) + ξ(d) ξ(c) + ξ(c + η(d, c)) ⊇ . ⊇ 2 2 Remark 6.3.1. When η(d, c) = d−c, then above theorem reduces to Theorem 3.4 of [25], i.e.   c+d Γ(α + 1) α ξ ⊇ [J + ξ(d) + Jdα− ξ(c)] 2 2(d − c)α c ξ(c) + ξ(d) ⊇ . (6.16) 2

156

Integral Inequalities and Generalized Convexity

We prove Hermite–Hadamard type inequalities for the product of two preinvex interval-valued functions [160]. Theorem 6.3.3. Let K ⊆ R be an open invex subset with respect to η : K × K → R and c, d ∈ K with c < c + η(d, c). If ξ, ψ : [c, c + η(d, c)] → R+ I is a preinvex interval-valued function such that ξ(δ) = [ξ(δ), ξ(δ)] and ψ(δ) = [ψ(δ), ψ(δ)]. ξ, ψ ∈ L1 [c, c + η(d, c)] and η satisfies Condition C and α > 0, then we have Γ(α + 1) α α [J + ξ(c + η(d, c))ψ(c + η(d, c)) + J(c+η(d,c)) − ξ(c)ψ(c)] 2η α (d, c) c   α 1 − M (c, c + η(d, c)) ⊇ 2 (α + 1)(α + 2) α + N (c, c + η(d, c)), (α + 1)(α + 2) where M (c, c + η(d, c)) = ξ(c)ψ(c) + ξ(c + η(d, c))ψ(c + η(d, c)) and N (c, c + η(d, c)) = ξ(c)ψ(c + η(d, c)) + ξ(c + η(d, c))ψ(c). Proof Since ξ and ψ are two preinvex interval-valued functions for δ ∈ [0, 1], we have ξ(c + δη(d, c)) = ξ(c + η(d, c) + (1 − δ)η(c, c + η(d, c))) ⊇ δξ(c + η(d, c)) + (1 − δ)ξ(c)

(6.17)

and ψ(c + δη(d, c)) = ψ(c + η(d, c) + (1 − δ)η(c, c + η(d, c))) ⊇ δψ(c + η(d, c)) + (1 − δ)ψ(c).

(6.18)

Since, ξ(x), ψ(x) ∈ R+ I , ∀ x ∈ [c, d], then from (6.17) and (6.18), we have ξ(c + δη(d, c))ψ(c + δη(d, c)) ⊇ δ 2 ξ(c + η(d, c))ψ(c + η(d, c)) + (1 − δ)2 ξ(c)ψ(c)

+ δ(1 − δ)[ξ(c + η(d, c))ψ(c) + ξ(c)ψ(c + η(d, c))].

(6.19)

Similarly, ξ(c + (1 − δ)η(d, c))ψ(c + (1 − δ)η(d, c))

⊇ δ 2 ξ(c)ψ(c) + (1 − δ)2 ξ(c + η(d, c))ψ(c + η(d, c))

+ δ(1 − δ)[ξ(c + η(d, c))ψ(c) + ξ(c)ψ(c + η(d, c))].

(6.20)

Hermite–Hadamard Type Inclusions for Interval-Valued Generalized...

157

Adding (6.19) and (6.20), we have ξ(c + δη(d, c))ψ(c + δη(d, c)) + ξ(c + (1 − δ)η(d, c))ψ(c + (1 − δ)η(d, c)) ⊇ δ 2 [ξ(c)ψ(c) + ξ(c + η(d, c))ψ(c + η(d, c))]

+ (1 − δ)2 [ξ(c)ψ(c) + ξ(c + η(d, c))ψ(c + η(d, c))] + 2δ(1 − δ)[ξ(c + η(d, c))ψ(c) + ξ(c)ψ(c + η(d, c))]

= [δ 2 + (1 − δ)2 ][ξ(c)ψ(c) + ξ(c + η(d, c))ψ(c + η(d, c))] + 2δ(1 − δ)[ξ(c + η(d, c))ψ(c) + ξ(c)ψ(c + η(d, c))]

= [2δ 2 − 2δ + 1]M (c, c + η(d, c)) + 2δ(1 − δ)N (c, c + η(d, c)).

(6.21)

Multiplying by δ α−1 on both sides and integrating on [0, 1], we have  1 δ α−1 ξ(c + δη(d, c))ψ(c + δη(d, c))dδ (IR) 0

+ (IR)



1

0

⊇ (IR) + (IR)



δ α−1 ξ(c + (1 − δ)η(d, c))ψ(c + (1 − δ)η(d, c))dδ



(IR)

[2δ α+1 − 2δ α + δ α−1 ]M (c, c + η(d, c))dδ

0

1

2[δ α − δ α+1 ]N (c, c + η(d, c))dδ.

0

Since,

1



(6.22)

1

δ α−1 ξ(c + δη(d, c))ψ(c + δη(d, c))dδ

0

=

(IR)



0

Γ(α) α J − ξ(c)ψ(c), η α (d, c) (c+η(d,c))

1

δ α−1 ξ(c + (1 − δ)η(d, c))ψ(c + (1 − δ)η(d, c))dδ Γ(α) α J + ξ(c α η (d, c) c

=

(IR)



(6.23)

+ η(d, c))ψ(c + η(d, c)),

(6.24)

1 0

[2δ α+1 − 2δ α + δ α−1 ]M (c, c + η(d, c))dδ   2 1 α = − M (c, c + η(d, c)) α 2 (α + 1)(α + 2)

and (IR)



0

=

(6.25)

1

2[δ α − δ α+1 ]N (c, c + η(d, c))dδ

2 N (c, c + η(d, c)). (α + 1)(α + 2)

(6.26)

158

Integral Inequalities and Generalized Convexity

Using (6.23), (6.24), (6.25) and (6.26) in (6.22), we have Γ(α) α [J α − ξ(c)ψ(c) + Jc+ ξ(c + η(d, c))ψ(c + η(d, c))] η α (d, c) (c+η(d,c))   α 2 1 − M (c, c + η(d, c)) ⊇ α 2 (α + 1)(α + 2) 2 N (c, c + η(d, c)). (6.27) + (α + 1)(α + 2) This implies, Γ(α + 1) α α [J + ξ(c + η(d, c))ψ(c + η(d, c)) + J(c+η(d,c)) − ξ(c)ψ(c)] 2η α (d, c) c   α 1 − M (c, c + η(d, c)) ⊇ 2 (α + 1)(α + 2) α + N (c, c + η(d, c)). (α + 1)(α + 2) Corollary 6.3.2. If α = 1, then above theorem reduces to the following result:  c+η(d,c) 1 1 1 ξ(δ)ψ(δ)dδ ⊇ M (c, c + η(d, c)) + N (c, c + η(d, c)). η(d, c) c 3 6 Remark 6.3.2. When η(d, c) = d−c, then above theorem reduces to Theorem 3.5 of [25], i.e. Γ(α + 1) α [J + ξ(d)ψ(d) + Jdα− ξ(c)ψ(c)] 2(d − c)α c   α α 1 − M (c, d) + N (c, d), ⊇ 2 (α + 1)(α + 2) (α + 1)(α + 2) where M (c, d) = ξ(c)ψ(c) + ξ(d)ψ(d) and N (c, d) = ξ(c)ψ(d) + ξ(d)ψ(c). Theorem 6.3.4. Let K ⊆ R be an open invex subset with respect to η : K × K → R and c, d ∈ K with c < c + η(d, c). If ξ, ψ : [c, c + η(d, c)] → R+ I is a preinvex interval-valued function such that ξ(δ) = [ξ(δ), ξ(δ)] and ψ(δ) = [ψ(δ), ψ(δ)]. ξ, ψ ∈ L1 [c, c + η(d, c)] and η satisfies Condition C and α > 0, then we have     1 1 2ξ c + η(d, c) ψ c + η(d, c) 2 2 Γ(α + 1) α α [J + ξ(c + η(d, c))ψ(c + η(d, c)) + J(c+η(d,c)) ⊇ α − ξ(c)ψ(c)] 2η (d, c) c   α α 1 − N (c, c + η(d, c)) + M (c, c + η(d, c)), + 2 (α + 1)(α + 2) (α + 1)(α + 2) where M (c, c + η(d, c)) and N (c, c + η(d, c)) are defined as previous.

Hermite–Hadamard Type Inclusions for Interval-Valued Generalized...

159

Proof Since ξ is a preinvex interval-valued function, we have   ξ(x) + ξ(y) 1 , ∀ x, y ∈ [c, c + η(d, c)]. ξ x + η(y, x) ⊇ 2 2 Using x = c + (1 − δ)η(d, c), y = c + δη(d, c) and Condition C in above, we get 1 ξ(c + (1 − δ)η(d, c) + η(c + δη(d, c), c + (1 − δ)η(d, c))) 2 ξ(c + (1 − δ)η(d, c)) + ξ(c + δη(d, c)) ⊇ . 2 This implies,   ξ(c + (1 − δ)η(d, c)) + ξ(c + δη(d, c)) 1 . ξ c + η(d, c) ⊇ 2 2 Similarly,   ψ(c + (1 − δ)η(d, c)) + ψ(c + δη(d, c)) 1 . ψ c + η(d, c) ⊇ 2 2

(6.28)

(6.29)

From (6.28) and (6.29), we get     1 1 ξ c + η(d, c) ψ c + η(d, c) 2 2 1 ⊇ [ξ(c + (1 − δ)η(d, c)) + ξ(c + δη(d, c))] 4 × [ψ(c + (1 − δ)η(d, c)) + ψ(c + δη(d, c))] 1 = [ξ(c + (1 − δ)η(d, c))ψ(c + (1 − δ)η(d, c)) 4 + ξ(c + δη(d, c))ψ(c + δη(d, c)) + ξ(c + (1 − δ)η(d, c))ψ(c + δη(d, c)) + ξ(c + δη(d, c))ψ(c + (1 − δ)η(d, c))].

(6.30)

Since ξ and ψ ∈ R+ I , ∀x ∈ [c, c + η(d, c)] are two preinvex interval-valued functions for δ ∈ [0, 1], we have ξ(c + (1 − δ)η(d, c))ψ(c + δη(d, c))

⊇ δ 2 ξ(c)ψ(c + η(d, c)) + (1 − δ)2 ξ(c + η(d, c))ψ(c)

+ δ(1 − δ)[ξ(c + η(d, c))ψ(c + η(d, c)) + ξ(c)ψ(c)].

(6.31)

Similarly, ξ(c + δη(d, c))ψ(c + (1 − δ)η(d, c))

⊇ δ 2 ξ(c + η(d, c))ψ(c) + (1 − δ)2 ξ(c)ψ(c + η(d, c))

+ δ(1 − δ)[ξ(c + η(d, c))ψ(c + η(d, c)) + ξ(c)ψ(c)].

(6.32)

160

Integral Inequalities and Generalized Convexity

Adding (6.31) and (6.32), we obtain ξ(c + (1 − δ)η(d, c))ψ(c + δη(d, c)) + ξ(c + δη(d, c))ψ(c + (1 − δ)η(d, c)) ⊇ [2δ 2 − 2δ + 1]N (c, c + η(d, c)) + 2δ(1 − δ)M (c, c + η(d, c)).

(6.33)

From (6.30) and (6.33), we have     1 1 1 ξ c + η(d, c) ψ c + η(d, c) ⊇ [(2δ 2 − 2δ + 1)N (c, c + η(d, c)) 2 2 4 1 + 2δ(1 − δ)M (c, c + η(d, c))] + [ξ(c + (1 − δ)η(d, c))ψ(c + (1 − δ)η(d, c)) 4 + ξ(c + δη(d, c))ψ(c + δη(d, c))]. Multiplying by δ α−1 on both sides in above, then integrating on [0, 1], we obtain     1  1 1 ξ c + η(d, c) ψ c + η(d, c) δ α−1 dδ (IR) 2 2 0  1 1 ⊇ (IR) δ α−1 (2δ 2 − 2δ + 1)N (c, c + η(d, c))dδ 4 0  1 1 + (IR) δ α (1 − δ)M (c, c + η(d, c))dδ 2 0  1 1 + (IR) ξ(c + (1 − δ)η(d, c))ψ(c + (1 − δ)η(d, c))δ α−1 dδ 4 0  1 1 + (IR) ξ(c + δη(d, c))ψ(c + δη(d, c))δ α−1 dδ. 4 0 This implies,     1 1 2ξ c + η(d, c) ψ c + η(d, c) 2 2 Γ(α + 1) α α [J + ξ(c + η(d, c))ψ(c + η(d, c)) + J(c+η(d,c)) ⊇ α − ξ(c)ψ(c)] 2η (d, c) c   α α 1 − N (c, c + η(d, c)) + M (c, c + η(d, c)). + 2 (α + 1)(α + 2) (α + 1)(α + 2) This completes the proof. Corollary 6.3.3. If α = 1, then Theorem (6.3.4) reduces to the following result:     1 1 2ξ c + η(d, c) ψ c + η(d, c) 2 2  c+η(d,c) 1 1 1 ⊇ ξ(δ)ψ(δ)dδ + N (c, c + η(d, c)) + M (c, c + η(d, c)). η(d, c) c 3 6

Hermite–Hadamard Type Inclusions for Interval-Valued Generalized...

161

Remark 6.3.3. When η(d, c) = d−c, then above theorem reduces to Theorem 3.6 of [25], i.e.     c+d c+d 2ξ ψ 2 2 Γ(α + 1) α [J + ξ(d)ψ(d) + Jdα− ξ(c)ψ(c)] ⊇ 2(d − c)α c   α α 1 − N (c, d) + M (c, d), + 2 (α + 1)(α + 2) (α + 1)(α + 2) where M (c, d) and N (c, d) are defined as previous.

6.4

Hermite–Hadamard Type Inclusions for Interval-Valued Coordinated Preinvex Functions

In this section, first, we give the definition of interval-valued coordinated preinvex function [93]. Definition 6.4.1. Let K1 × K2 be an invex set with respect to η1 and η2 , ξ = [ξ, ξ] be an interval valued function defined on K1 × K2 . The function ξ is said to be interval-valued coordinated preinvex function with respect to η1 and η2 if the partial mappings ξv : K1 → R+ I , ξv (w) = ξ(w, v) and ξu : K2 → , ξ (z) = ξ(u, z) are interval-valued preinvex functions with respect to η1 R+ u I and η2 , respectively, for all u ∈ K1 and v ∈ K2 . Remark 6.4.1. From the definition of interval-valued coordinated preinvex functions, it follows that if ξ is an interval-valued coordinated preinvex function, then ξ(u + δ1 η1 (w, u), v + δ2 η2 (z, v)) ⊇ (1 − δ1 )(1 − δ2 )ξ(u, v) + (1 − δ1 )δ2 ξ(u, z) + δ1 (1 − δ2 )ξ(w, v) + δ1 δ2 ξ(w, z), for all (u, v), (u, z), (w, v), (w, z) ∈ K1 × K2 and δ1 , δ2 ∈ [0, 1]. If η1 (w, u) = w − u and η2 (z, v) = z − v, then the definition of intervalvalued coordinated preinvex function reduces to the definition of intervalvalued coordinated convex function proposed by Zhao et al. [183]. Example 6.4.1. An interval-valued function ξ : [0, 1] × [ 12 , 1] → R+ I defined as ξ(u, v) = [u + v, (2 − u)(2 − v)] is an interval-valued coordinated preinvex function with respect to η1 (w,  u) = w − u − 1 and η2 (z, v) = z − 2v for all u, w ∈ [0, 1] and v, z ∈ 12 , 1 .

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Integral Inequalities and Generalized Convexity

Now, we establish Hermite–Hadamard type inclusions for interval-valued preinvex functions on coordinates [93]. Throughout this section, we will not include the symbols (R), (IR) and (ID) before the integral sign. Theorem 6.4.1. Let K1 × K2 be an invex set with respect to η1 and η2 . If ξ : K 1 × K 2 → R+ I is an interval-valued coordinated preinvex function with respect to η1 and η2 such that ξ = [ξ, ξ] and a < a + η1 (b, a), c < c + η2 (d, c), where a, b ∈ K1 and c, d ∈ K2 . If η1 , η2 satisfy Condition C, then we have 

1 1 ξ a + η1 (b, a), c + η2 (d, c) 2 2 1 ⊇ η1 (b, a)η2 (d, c) ⊇



a+η1 (b,a)

a





c+η2 (d,c)

ξ(u, v)dvdu c

1 [ξ(a, c) + ξ(b, c) + ξ(a, d) + ξ(b, d)]. 4

Proof Since ξ is an interval-valued preinvex function on coordinates with respect to η1 and η2 , we have ξ(a + δ1 η1 (b, a), c + δ2 η2 (d, c)) ⊇ (1 − δ1 )(1 − δ2 )ξ(a, c) + (1 − δ1 )δ2 ξ(a, d) + δ1 (1 − δ2 )ξ(b, c) + δ1 δ2 ξ(b, d).

(6.34)

Integrating (6.34) with respect to (δ1 , δ2 ) over [0, 1] × [0, 1], we get 

1 0

⊇





1

ξ(a + δ1 η1 (b, a), c + δ2 η2 (d, c))dδ2 dδ1 0

1 0

+





1 0

1 0



(1 − δ1 )(1 − δ2 )ξ(a, c)dδ2 dδ1 + 1 0

δ1 (1 − δ2 )ξ(b, c)dδ2 dδ1 +



1 0



1 0





1 0

(1 − δ1 )δ2 ξ(a, d)dδ2 dδ1

1

δ1 δ2 ξ(b, d)dδ2 dδ1 . 0

This implies that  a+η1 (b,a)  c+η2 (d,c) 1 ξ(u, v)dvdu η1 (b, a)η2 (d, c) a c 1 ⊇ [ξ(a, c) + ξ(a, d) + ξ(b, c) + ξ(b, d)]. 4

(6.35)

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163

Using the definition of an interval-valued coordinated preinvex function and Condition C for η1 , η2 , we get   1 1 ξ a + η1 (b, a), c + η2 (d, c) 2 2 1 = ξ(a + δ1 η1 (b, a) + η1 (a + (1 − δ1 )η1 (b, a), a + δ1 η1 (b, a)), c + δ2 η2 (d, c) 2 1 + η2 (c + (1 − δ2 )η2 (d, c), c + δ2 η2 (d, c))) 2 1 ⊇ [ξ(a + δ1 η1 (b, a), c + δ2 η2 (d, c)) 4 + ξ(a + δ1 η1 (b, a), c + (1 − δ2 )η2 (d, c)) + ξ(a + (1 − δ1 )η1 (b, a), c + δ2 η2 (d, c)) + ξ(a + (1 − δ1 )η1 (b, a), c + (1 − δ2 )η2 (d, c))]

(6.36)

Thus, integrating (6.36) with respect to (δ1 , δ2 ) over [0, 1] × [0, 1], we get   1 1 ξ a + η1 (b, a), c + η2 (d, c) dδ2 dδ1 2 2 0 0  1 1 1 ⊇ [ξ(a + δ1 η1 (b, a), c + δ2 η2 (d, c)) 4 0 0 + ξ(a + δ1 η1 (b, a), c + (1 − δ2 )η2 (d, c)) + ξ(a + (1 − δ1 )η1 (b, a), c + δ2 η2 (d, c))



1



1

+ ξ(a + (1 − δ1 )η1 (b, a), c + (1 − δ2 )η2 (d, c))]dδ2 dδ1 .

This implies   1 1 ξ a + η1 (b, a), c + η2 (d, c) 2 2  a+η1 (b,a)  c+η2 (d,c) 1 ξ(u, v)dvdu. ⊇ η1 (b, a)η2 (d, c) a c

(6.37)

From (6.35) and (6.37), we get the desired result. Theorem 6.4.2. Let K1 × K2 be an invex set with respect to η1 and η2 . If ξ : [a, a + η1 (b, a)] × [c, c + η2 (d, c)] → R+ I is an interval-valued coordinated preinvex function with respect to η1 and η2 such that ξ = [ξ, ξ] and a < a + η1 (b, a), c < c + η2 (d, c), where a, b ∈ K1 and c, d ∈ K2 . If η1 , η2 satisfy

164

Integral Inequalities and Generalized Convexity

Condition C, then we have   1 ξ u, c + η2 (d, c)) du 2 a   c+η2 (d,c)  1 1 ξ a + η1 (b, a), v dv + η2 (d, c) c 2

1 η1 (b, a)



a+η1 (b,a)

 a+η1 (b,a)  c+η2 (d,c) 2 ξ(u, v)dvdu η1 (b, a)η2 (d, c) a c   a+η1 (b,a) 1 1 (ξ(u, c) + ξ(u, c + η2 (d, c)))du ⊇ 2 η1 (b, a) a

⊇



1 + η2 (d, c)

c+η2 (d,c)



(ξ(a, v) + ξ(a + η1 (b, a), v))dv .

c

(6.38)

Proof Since ξ is an interval-valued preinvex function on coordinates [a, a + η1 (b, a)] × [c, c + η2 (d, c)], then ξu : [c, c + η2 (d, c)] → R+ I , ξu (v) = ξ(u, v) is an interval-valued preinvex function on [c, c + η2 (d, c)] for all u ∈ [a, a + η1 (b, a)]. From Corollary 6.3.1, we have    c+η2 (d,c) 1 1 ξu (r) + ξu (c + η2 (d, c)) ξu c + η2 (d, c)) ⊇ . ξu (v)dv ⊇ 2 η2 (d, c) c 2 This implies    c+η2 (d,c) 1 1 ξ u, c + η2 (d, c)) ⊇ ξ(u, v)dv 2 η2 (d, c) c ⊇

ξ(u, c) + ξ(u, c + η2 (d, c)) . 2

(6.39)

Integrating (6.39) over [a, a + η1 (b, a)] with respect to u, then dividing by η1 (b, a), we get 1 η1 (b, a)



a+η1 (b,a)

a

1 ⊇ η1 (b, a)η2 (d, c) 1 ⊇ 2η1 (b, a)



  1 ξ u, c + η2 (d, c)) du 2



a+η1 (b,a)

a



c+η2 (d,c)

ξ(u, v)dvdu c

a+η1 (b,a)

(ξ(u, c) + ξ(u, c + η2 (d, c)))du. a

(6.40)

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165

Similarly, ξv : [a, a + η1 (a, b)] → R+ I , ξv (u) = ξ(u, v) is interval-valued preinvex function on [a, a + η1 (a, b)] for all v ∈ [c, c + η2 (d, c)]. Then, we have 1 η2 (d, c) ⊇



c+η2 (d,c)

c

1 η1 (b, a)η2 (d, c)

1 ⊇ 2η2 (d, c)



  1 ξ a + η1 (b, a), v dv 2



a+η1 (b,a)

a



c+η2 (d,c)

ξ(u, v)dvdu c

c+η2 (d,c)

(ξ(a, v) + ξ(a + η1 (b, a), v))dv.

(6.41)

c

By adding (6.40) and (6.41), we have   1 ξ u, c + η2 (d, c)) du 2 a   c+η2 (d,c)  1 1 ξ a + η1 (b, a), v dv + η2 (d, c) c 2

1 η1 (b, a)



a+η1 (b,a)

 a+η1 (b,a)  c+η2 (d,c) 2 ξ(u, v)dvdu η1 (b, a)η2 (d, c) a c   a+η1 (b,a) 1 1 (ξ(u, c) + ξ(u, c + η2 (d, c)))du ⊇ 2 η1 (b, a) a ⊇

1 + η2 (d, c)



c+η2 (d,c)

c



(ξ(a, v) + ξ(a + η1 (b, a), v))dv .

This completes the proof. Example 6.4.2. Let [a, a + η1 (b, a)] = [ 14 , 12 ], [c, c + η2 (d, c)] = [ 14 , 12 ] and η1 (b, a) = b − 2a, η2 (d, c) = d − 2c. Let ξ : [ 14 , 12 ] × [ 14 , 12 ] → R+ I be defined by ξ(u, v) = [uv, (1 − u)(1 − v)] ∀ u ∈ [ 41 , 12 ] and v ∈ [ 14 , 12 ]. Then all assumptions of Theorem 6.4.2 are satisfied. Theorem 6.4.3. Let K1 × K2 be an invex set with respect to η1 and η2 . If ξ : [a, a + η1 (b, a)] × [c, c + η2 (d, c)] → R+ I is an interval-valued coordinated preinvex function with respect to η1 and η2 such that ξ = [ξ, ξ] and a < a + η1 (b, a), c < c + η2 (d, c), where a, b ∈ K1 and c, d ∈ K2 . If η1 , η2 satisfy

166

Integral Inequalities and Generalized Convexity

Condition C, then we have   1 1 ξ a + η1 (b, a), c + η2 (d, c)) 2 2    a+η1 (b,a)  1 1 1 ⊇ ξ u, c + η2 (d, c)) du 2 η1 (b, a) a 2 1 + η2 (d, c)



c+η2 (d,c)

c

   1 ξ a + η1 (b, a), v dv 2

 a+η1 (b,a)  c+η2 (d,c) 1 ξ(u, v)dvdu ⊇ η1 (b, a)η2 (d, c) a c   a+η1 (b,a) 1 1 (ξ(u, c) + ξ(u, c + η2 (d, c)))du ⊇ 4 η1 (b, a) a 1 + η2 (d, c) ⊇



c+η2 (d,c)

(ξ(a, v) + ξ(a + η1 (b, a), v))dv

c



1 [ξ(a, c) + ξ(a + η1 (b, a), c) + ξ(a, c + η2 (d, c)) 4 + ξ(a + η1 (b, a), c + η2 (d, c))]

⊇

1 [ξ(a, c) + ξ(b, c) + ξ(a, d) + ξ(b, d)]. 4

Proof Since ξ is an interval-valued preinvex function on coordinates [a, a + η1 (b, a)] × [c, c + η2 (d, c)], then from Corollary 6.3.1 we get   1 1 ξ a + η1 (b, a), c + η2 (d, c)) 2 2   a+η1 (b,a)  1 1 ξ u, c + η2 (d, c)) du, (6.42) ⊇ η1 (b, a) a 2 

 1 1 ξ a + η1 (b, a), c + η2 (d, c)) 2 2   c+η2 (d,c)  1 1 ξ a + η1 (b, a), v dv. ⊇ η2 (d, c) c 2

(6.43)

Hermite–Hadamard Type Inclusions for Interval-Valued Generalized... Adding (6.42) and (6.43), we have   1 1 ξ a + η1 (b, a), c + η2 (d, c)) 2 2    a+η1 (b,a)  1 1 1 ⊇ ξ u, c + η2 (d, c)) du 2 η1 (b, a) a 2    c+η2 (d,c)  1 1 + ξ a + η1 (b, a), v dv . η2 (d, c) c 2 Again from Corollary 6.3.1, we get  a+η1 (b,a) ξ(a, c) + ξ(a + η1 (b, a), c) 1 , ξ(u, c)du ⊇ η1 (b, a) a 2  a+η1 (b,a) 1 ξ(u, c + η2 (d, c))du η1 (b, a) a ξ(a, c + η2 (d, c)) + ξ(a + η1 (b, a), c + η2 (d, c)) ⊇ , 2 1 η2 (d, c)



c+η2 (d,c)

c

ξ(a, v)dv ⊇

ξ(a, c) + ξ(a, c + η2 (d, c)) , 2

 c+η2 (d,c) 1 ξ(a + η1 (b, a), v)dv η2 (d, c) c ξ(a + η1 (b, a), c) + ξ(a + η1 (b, a), c + η2 (d, c)) ⊇ . 2 Adding (6.45)–(6.48), we get  a+η1 (b,a) 1 (ξ(u, c) + ξ(u, c + η2 (d, c)))du η1 (b, a) a  c+η2 (d,c) 1 (ξ(a, v) + ξ(a + η1 (b, a), v))dv + η2 (d, c) c ⊇ ξ(a, c) + ξ(a + η1 (b, a), c) + ξ(a, c + η2 (d, c)) + ξ(a + η1 (b, a), c + η2 (d, c)).

167

(6.44)

(6.45)

(6.46)

(6.47)

(6.48)

(6.49)

By Corollary 6.3.1, we also have ξ(a, c) + ξ(a + η1 (b, a), c) + ξ(a, c + η2 (d, c)) + ξ(a + η1 (b, a), c + η2 (d, c)) ⊇ ξ(a, c) + ξ(b, c) + ξ(a, d) + ξ(b, d). From (6.38), (6.44), (6.49), and (6.50), we get the desired result.

(6.50)

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Integral Inequalities and Generalized Convexity

Remark 6.4.2. If we put η1 (b, a) = b − a and η2 (d, c) = d − c in Theorem 6.4.3, we obtain Theorem 7 of [183], i.e.   a+b c+d ξ , 2 2      b   d  1 1 1 c+d a+b du + , v dv ⊇ ξ u, ξ 2 (b − a) a 2 (d − c) c 2  b d 1 ⊇ ξ(u, v)dvdu (b − a)(d − c) a c   b 1 1 ⊇ (ξ(u, c) + ξ(u, d))du 4 (b − a) a   d 1 + (ξ(a, v) + ξ(b, v))dv (d − c) c ⊇

1 [ξ(a, c) + ξ(b, c) + ξ(a, d) + ξ(b, d)]. 4

Next, we prove Hermite–Hadamard type inclusions for the product of two interval-valued coordinated preinvex functions [93]. Theorem 6.4.4. Let K1 × K2 be an invex set with respect to η1 and η2 . If ξ, ψ : [a, a + η1 (b, a)] × [c, c + η2 (d, c)] → R+ I are interval-valued coordinated preinvex functions with respect to η1 and η2 such that ξ = [ξ, ξ], ψ = [ψ, ψ] and a < a + η1 (b, a), c < c + η2 (d, c), where a, b ∈ K1 and c, d ∈ K2 . If η1 , η2 satisfy Condition C, then  a+η1 (b,a)  c+η2 (d,c) 1 ξ(u, v)ψ(u, v)dvdu η1 (b, a)η2 (d, c) a c 1 1 1 1 ⊇ N1 (a, b, c, d) + N2 (a, b, c, d) + N3 (a, b, c, d) + N4 (a, b, c, d), 9 18 18 36 where N1 (a, b, c, d) = ξ(a, c)ψ(a, c) + ξ(a + η1 (b, a), c)ψ(a + η1 (b, a), c) + ξ(a, c + η2 (d, c))ψ(a, c+η2 (d, c))+ξ(a+η1 (b, a), c+η2 (d, c))ψ(a+η1 (b, a), c+η2 (d, c)), N2 (a, b, c, d) = ξ(a, c)ψ(a + η1 (b, a), c) + ξ(a + η1 (b, a), c)ψ(a, c) + ξ(a, c + η2 (d, c))ψ(a+η1 (b, a), c+η2 (d, c))+ξ(a+η1 (b, a), c+η2 (d, c))ψ(a, c+η2 (d, c)), N3 (a, b, c, d) = ξ(a, c)ψ(a, c + η2 (d, c)) + ξ(a + η1 (b, a), c)ψ(a + η1 (b, a), c + η2 (d, c)) + ξ(a, c + η2 (d, c))ψ(a, c) + ξ(a + η1 (b, a), c + η2 (d, c))ψ(a + η1 (b, a), c), N4 (a, b, c, d) = ξ(a, c)ψ(a + η1 (b, a), c + η2 (d, c)) + ξ(a + η1 (b, a), c)ψ(a, c + η2 (d, c)) + ξ(a, c + η2 (d, c))ψ(a + η1 (b, a), c) + ξ(a + η1 (b, a), c + η2 (d, c))ψ(a, c).

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169

Proof Since ξ and ψ are interval-valued coordinated preinvex functions on [a, a + η1 (b, a)] × [c, c + η2 (d, c)], we have and

ξu (v) : [c, c + η2 (d, c)] → R+ I , ξu (v) = ξ(u, v)

ψu (v) : [c, c + η2 (d, c)] → R+ I , ψu (v) = ψ(u, v) are interval-valued preinvex functions on [c, c + η2 (d, c)] for all u ∈ [a, a + η1 (b, a)]. Similarly, ξv (u) : [a, a + η1 (b, a)] → R+ I , ξv (u) = ξ(u, v) and ψv (u) : [a, a + η1 (b, a)] → R+ I , ψv (u) = ψ(u, v) are interval-valued preinvex functions on [a, a + η1 (b, a)] for all v ∈ [c, c + η2 (d, c)]. From Corollary (6.3.2), we get  c+η2 (d,c) 1 ξu (v)ψu (v)dv η2 (d, c) c 1 ⊇ [ξu (c)ψu (c) + ξu (c + η2 (d, c))ψu (c + η2 (d, c))] 3 1 + [ξu (c)ψu (c + η2 (d, c)) + ξu (c + η2 (d, c))ψu (c)]. 6 This implies  c+η2 (d,c) 1 ξ(u, v)ψ(u, v)dv η2 (d, c) c 1 ⊇ [ξ(u, c)ψ(u, c) + ξ(u, c + η2 (d, c))ψ(u, c + η2 (d, c))] 3 1 + [ξ(u, c)ψ(u, c + η2 (d, c)) + ξ(u, c + η2 (d, c))ψ(u, c)]. 6

(6.51)

Integrating (6.51) with respect to u over [a, a + η1 (b, a)] and after then dividing by η1 (b, a), we find  a+η1 (b,a)  c+η2 (d,c) 1 ξ(u, v)ψ(u, v)dvdu η1 (b, a)η2 (d, c) a c  a+η1 (b,a) 1 [ξ(u, c)ψ(u, c) + ξ(u, c + η2 (d, c))ψ(u, c + η2 (d, c))]du ⊇ 3η1 (b, a) a  a+η1 (b,a) 1 [ξ(u, c)ψ(u, c + η2 (d, c)) + ξ(u, c + η2 (d, c))ψ(u, c)]du. + 6η1 (b, a) a (6.52)

170

Integral Inequalities and Generalized Convexity Again from Corollary 6.3.2, we have  a+η1 (b,a) 1 ξ(u, c)ψ(u, c)du η1 (b, a) a 1 ⊇ [ξ(a, c)ψ(a, c) + ξ(a + η1 (b, a), c)ψ(a + η1 (b, a), c] 3 1 + [ξ(a, c)ψ(a + η1 (b, a), c) + ξ(a + η1 (b, a), c)ψ(a, c)], 6  a+η1 (b,a) 1 ξ(u, c + η2 (d, c))ψ(u, c + η2 (d, c))du η1 (b, a) a 1 ⊇ [ξ(a, c + η2 (d, c))ψ(a, c + η2 (d, c)) 3 + ξ(a + η1 (b, a), c + η2 (d, c))ψ(a + η1 (b, a), c + η2 (d, c))] 1 + [ξ(a, c + η2 (d, c))ψ(a + η1 (b, a), c + η2 (d, c)) 6 + ξ(a + η1 (b, a), c + η2 (d, c))ψ(a, c + η2 (d, c))],

(6.53)

(6.54)

 a+η1 (b,a) 1 ξ(u, c)ψ(u, c + η2 (d, c))du η1 (b, a) a 1 ⊇ [ξ(a, c))ψ(a, c + η2 (d, c)) + ξ(a + η1 (b, a), c)ψ(a + η1 (b, a), c + η2 (d, c))] 3 1 + [ξ(a, c)ψ(a + η1 (b, a), c + η2 (d, c)) + ξ(a + η1 (b, a), r)ψ(a, c + η2 (d, c))], 6 (6.55)  a+η1 (b,a) 1 ξ(u, c + η2 (d, c))ψ(u, c)du η1 (b, a) a 1 ⊇ [ξ(a, c + η2 (d, c))ψ(a, c) + ξ(a + η1 (b, a), c + η2 (d, c))ψ(a + η1 (b, a), c] 3 1 + [ξ(a, c + η2 (d, c))ψ(a + η1 (b, a), c) + ξ(p + η1 (b, a), c + η2 (d, c))ψ(a, c)]. 6 (6.56) Substituting (6.53)–(6.56) into (6.52), we obtain the desired result. Similarly, we can obtain the same result by using Corollary 6.3.2 for the product ξv (u)ψv (u) on [a, a + η1 (b, a)].

Hermite–Hadamard Type Inclusions for Interval-Valued Generalized...

171

Remark 6.4.3. If we put η1 (b, a) = b − a and η2 (d, c) = d − c in Theorem 6.4.4, we obtain Theorem 8 of [183], i.e.,  b d 1 ξ(u, v)ψ(u, v)dvdu (b − a)(d − c) a c 1 1 1 ⊇ N1∗ (a, b, c, d) + N2∗ (a, b, c, d) + N3∗ (a, b, c, d) 9 18 18 1 + N4∗ (a, b, c, d), 36 where N1∗ (a, b, c, d) = ξ(a, c)ψ(a, c) + ξ(b, c)ψ(b, c) + ξ(a, d) ψ(a, d) + ξ(b, d)ψ(b, d), N2∗ (a, b, c, d) = ξ(a, c)ψ(b, c) + ξ(b, c)ψ(a, c) + ξ(a, d) ψ(b, d)) + ξ(b, d)ψ(a, d), N3∗ (a, b, c, d) = ξ(a, c)ψ(a, d) + ξ(b, c)ψ(b, d) + ξ(a, d)ψ(a, c) + ξ(b, d)ψ(b, c), N4∗ (a, b, c, d) = ξ(a, c)ψ(b, d) + ξ(b, c)ψ(a, d) + ξ(a, d)ψ(b, c) + ξ(b, d)ψ(a, c). Theorem 6.4.5. Let K1 × K2 be an invex set with respect to η1 and η2 . If ξ, ψ : [a, a + η1 (b, a)] × [c, c + η2 (d, c)] → R+ I are interval-valued coordinated preinvex functions with respect to η1 and η2 such that ξ = [ξ, ξ], ψ = [ψ, ψ] and a < a + η1 (b, a), c < c + η2 (d, c), where a, b ∈ K1 and c, d ∈ K2 . If η1 , η2 satisfy Condition C, then we have     1 1 1 1 4ξ a + η1 (b, a), c + η2 (d, c) ψ a + η1 (b, a), c + η2 (d, c) 2 2 2 2  a+η1 (b,a)  c+η2 (d,c) 1 ξ(u, v)ψ(u, v)dvdu ⊇ η1 (b, a)η2 (d, c) a c 7 7 2 5 + N1 (a, b, c, d) + N2 (a, b, c, d) + N3 (a, b, c, d) + N4 (a, b, c, d), 36 36 36 9 where N1 (a, b, c, d), N2 (a, b, c, d), N3 (a, b, c, d), and N4 (a, b, c, d) are defined as previous. Proof Since ξ and ψ are interval-valued coordinated preinvex functions, therefore from Corollary 6.3.3, we have     1 1 1 1 2ξ a + η1 (b, a), c + η2 (d, c) ψ a + η1 (b, a), c + η2 (d, c) 2 2 2 2  a+η1 (b,a) 1 1 1 ξ(u, c + η2 (d, c))ψ(u, c + η2 (d, c))du ⊇ η1 (b, a) a 2 2  1 1 1 ξ(a, c + η2 (d, c))ψ(a, c + η2 (d, c)) + 6 2 2  1 1 +ξ(a + η1 (b, a), c + η2 (d, c))ψ(a + η1 (b, a), c + η2 (d, c)) 2 2

172

Integral Inequalities and Generalized Convexity  1 1 1 ξ(a, c + η2 (d, c))ψ(a + η1 (b, a), c + η2 (d, c)) + 3 2 2  1 1 +ξ(a + η1 (b, a), c + η2 (d, c))ψ(a, c + η2 (d, c)) 2 2

(6.57)

and     1 1 1 1 2ξ a + η1 (b, a), c + η2 (d, c) ψ a + η1 (b, a), c + η2 (d, c) 2 2 2 2  c+η2 (d,c) 1 1 1 ξ(c + η2 (d, c), v)ψ(a + η1 (b, a), v)dv ⊇ η2 (d, c) c 2 2  1 1 1 ξ(a + η1 (b, a), c)ψ(a + η1 (b, a), c) + 6 2 2  1 1 +ξ(a + η1 (b, a), c + η2 (d, c))ψ(a + η1 (b, a), c + η2 (d, c)) 2 2  1 1 1 ξ(a + η1 (b, a), c)ψ(a + η1 (b, a), c + η2 (d, c)) + 3 2 2  1 1 +ξ(a + η1 (b, a), c + η2 (d, c))ψ(a + η1 (b, a), c) . (6.58) 2 2 Adding (6.57) and (6.58), then multiplying both sides of the resultant one by two, we find     1 1 1 1 8ξ a + η1 (b, a), c + η2 (d, c) ψ a + η1 (b, a), c + η2 (d, c) 2 2 2 2  a+η1 (b,a) 1 1 2 ξ(u, c + η2 (d, c))ψ(u, c + η2 (d, c))du ⊇ η1 (b, a) a 2 2  c+η2 (d,c) 1 2 1 ξ(c + η2 (d, c), v)ψ(a + η1 (b, a), v)dv + η2 (d, c) c 2 2  1 1 1 2ξ(a, c + η2 (d, c))ψ(a, c + η2 (d, c)) + 6 2 2 1 1 +2ξ(a + η1 (b, a), c + η2 (d, c))ψ(a + η1 (b, a), c + η2 (d, c)) 2 2 1 1 +2ξ(a + η1 (b, a), c)ψ(a + η1 (b, a), c) 2 2  1 1 +2ξ(a + η1 (b, a), c + η2 (d, c))ψ(a + η1 (b, a), c + η2 (d, c)) 2 2  1 1 1 + 2ξ(a, c + η2 (d, c))ψ(a + η1 (b, a), c + η2 (d, c)) 3 2 2 1 1 + 2ξ(a + η1 (b, a), c + η2 (d, c))ψ(a, c + η2 (d, c)) 2 2

Hermite–Hadamard Type Inclusions for Interval-Valued Generalized...

173

1 1 + 2ξ(a + η1 (b, a), c)ψ(a + η1 (b, a), c + η2 (d, c)) 2 2  1 1 + 2ξ(a + η1 (b, a), c + η2 (d, c))ψ(a + η1 (b, a), c) . 2 2

(6.59)

Now, from Corollary 6.3.3, we have 1 1 2ξ(a, c + η2 (d, c))ψ(a, c + η2 (d, c)) 2 2  c+η2 (d,c) 1 ξ(a, v)ψ(a, v)dv ⊇ η2 (d, c) c 1 + [ξ(a, c)ψ(a, c) + ξ(a, c + η2 (d, c))ψ(a, c + η2 (d, c))] 6 1 + [ξ(a, c)ψ(a, c + η2 (d, c)) + ξ(a, c + η2 (d, c))ψ(a, c)], 3

(6.60)

1 1 2ξ(a + η1 (b, a), c + η2 (d, c))ψ(a + η1 (b, a), c + η2 (d, c)) 2 2  c+η2 (d,c) 1 ξ(a + η1 (b, a), v)ψ(a + η1 (b, a), v)dv ⊇ η2 (d, c) c +

1 [ξ(a + η1 (b, a), c)ψ(a + η1 (b, a), c) 6

+ξ(a + η1 (b, a), c + η2 (d, c))ψ(a + η1 (b, a), c + η2 (d, c))] +

1 [ξ(a + η1 (b, a), c)ψ(a + η1 (b, a), c + η2 (d, c)) 3

+ξ(a + η1 (b, a), c + η2 (d, c))ψ(a + η1 (b, a), c)] ,

(6.61)

    1 1 2ξ a + η1 (b, a), c ψ a + η1 (b, a), c 2 2 ⊇

1 η1 (b, a)



a+η1 (b,a)

ξ(u, c)ψ(u, c)du

a

1 + [ξ(a, c)ψ(a, c) + ξ(a + η1 (b, a), c)ψ(a + η1 (b, a), c)] 6 1 + [ξ(a, c)ψ(a + η1 (b, a), c) + ξ(a + η1 (b, a), c)ψ(a, c)], 3

(6.62)

174

Integral Inequalities and Generalized Convexity     1 1 2ξ a + η1 (b, a), c + η2 (d, c) ψ a + η1 (b, a), c + η2 (d, c) 2 2  a+η1 (b,a) 1 ξ(u, c + η2 (d, c))ψ(u, c + η2 (d, c))du ⊇ η1 (b, a) a 1 + [ξ(a, c + η2 (d, c))ψ(a, c + η2 (d, c)) 6 + ξ(a + η1 (b, a), c + η2 (d, c))ψ(a + η1 (b, a), c + η2 (d, c))] 1 + [ξ(a, c + η2 (d, c))ψ(a + η1 (b, a), c + η2 (d, c)) 3 + ξ(a + η1 (b, a), c + η2 (d, c))ψ(a, c + η2 (d, c))],

(6.63)

1 1 2ξ(a, c + η2 (d, c))ψ(a + η1 (b, a), c + η2 (d, c)) 2 2  c+η2 (d,c) 1 ξ(a, v)ψ(a + η1 (b, a), v)dv ⊇ η2 (d, c) c 1 + [ξ(a, c)ψ(a + η1 (b, a), c) + ξ(a, c + η2 (d, c))ψ(a + η1 (b, a), c + η2 (d, c))] 6 1 + [ξ(a, c)ψ(a + η1 (b, a), c + η2 (d, c)) + ξ(a, c + η2 (d, c))ψ(a + η1 (b, a), c)], 3 (6.64) 1 1 2ξ(a + η1 (b, a), c + η2 (d, c))ψ(a, c + η2 (d, c)) 2 2  c+η2 (d,c) 1 ξ(a + η1 (b, a), v)ψ(a, v)dv ⊇ η2 (d, c) c 1 + [ξ(a + η1 (b, a), c)ψ(a, c) + ξ(a + η1 (b, a), c + η2 (d, c))ψ(a, c + η2 (d, c))] 6 1 + [ξ(a + η1 (b, a), c)ψ(a, c + η2 (d, c)) + ξ(a + η1 (b, a), c + η2 (d, c))ψ(a, c)], 3 (6.65)    1 1 2ξ a + η1 (b, a), c ψ a + η1 (b, a), c + η2 (d, c) 2 2  a+η1 (b,a) 1 ξ(u, c)ψ(u, c + η2 (d, c))du ⊇ η1 (b, a) a 1 + [ξ(a, c)ψ(a, c + η2 (d, c)) + ξ(a + η1 (b, a), c)ψ(a + η1 (b, a), c + η2 (d, c))] 6 1 + [ξ(a, c)ψ(a + η1 (b, a), c + η2 (d, c)) + ξ(a + η1 (b, a), c)ψ(a, c + η2 (d, c))], 3 (6.66) 

Hermite–Hadamard Type Inclusions for Interval-Valued Generalized...     1 1 2ξ a + η1 (b, a), c + η2 (d, c) ψ a + η1 (b, a), c 2 2  a+η1 (b,a) 1 ξ(u, c + η2 (d, c))ψ(u, c)du ⊇ η1 (b, a) a

175

1 + [ξ(a, c + η2 (d, c))ψ(a, c) + ξ(a + η1 (b, a), c + η2 (d, c))ψ(a + η1 (b, a), c)] 6 1 + [ξ(a, c + η2 (d, c))ψ(a + η1 (b, a), c) + ξ(a + η1 (b, a), c + η2 (d, c))ψ(a, c)]. 3 (6.67) Using (6.60)–(6.67) in (6.59), we get     1 1 1 1 8ξ a + η1 (b, a), c + η2 (d, c) ψ a + η1 (b, a), c + η2 (d, c) 2 2 2 2  a+η1 (b,a) 1 1 2 ξ(u, c + η2 (d, c))ψ(u, c + η2 (d, c))du ⊇ η1 (b, a) a 2 2  c+η2 (d,c) 1 1 2 ξ(c + η2 (d, c), v)ψ(a + η1 (b, a), v)dv + η2 (d, c) c 2 2  c+η2 (d,c) 1 (ξ(a, v)ψ(a, v) + ξ(a + η1 (b, a), v)ψ(a + η1 (b, a), v))dv + 6η2 (d, c) c  c+η2 (d,c) 1 (ξ(a, v)ψ(a + η1 (b, a), v) + ξ(a + η1 (b, a), v)ψ(a, v))dv + 3η2 (d, c) c  a+η1 (b,a) 1 (ξ(u, c)ψ(u, c) + ξ(u, c + η2 (d, c))ψ(u, c + η2 (d, c)))du + 6η1 (b, a) a  a+η1 (b,a) 1 (ξ(u, c)ψ(u, c + η2 (d, c)) + ξ(u, c + η2 (d, c))ψ(u, c))du + 3η1 (b, a) a 1 1 2 1 + N1 (a, b, c, d) + N2 (a, b, c, d) + N3 (a, b, c, d) + N4 (a, b, c, d). (6.68) 18 9 9 9 Again from Corollary 6.3.3, we have  c+η2 (d,c) 1 1 2 ξ(a + η1 (b, a), v)ψ(a + η1 (b, a), v)dv η2 (d, c) c 2 2  a+η1 (b,a)  c+η2 (d,c) 1 ξ(u, v)ψ(u, v)dvdu ⊇ η1 (b, a)η2 (d, c) a c  c+η2 (d,c) 1 (ξ(a, v)ψ(a, v) + ξ(a + η1 (b, a), v)ψ(a + η1 (b, a), v))dv + 6η2 (d, c) c  c+η2 (d,c) 1 (ξ(a, v)ψ(a + η1 (b, a), v) + ξ(a + η1 (b, a), v)ψ(a, v))dv, + 3η2 (d, c) c (6.69)

176

Integral Inequalities and Generalized Convexity  a+η1 (b,a) 2 1 1 ξ(u, c + η2 (d, c))ψ(u, c + η2 (d, c))du η1 (b, a) a 2 2  a+η1 (b,a)  c+η2 (d,c) 1 ξ(u, v)ψ(u, v)dvdu ⊇ η1 (b, a)η2 (d, c) a c

1 + 6η1 (b, a) +

1 3η1 (b, a)



a+η1 (b,a)

(ξ(u, c)ψ(u, c) + ξ(u, c + η2 (d, c))ψ(u, c + η2 (d, c)))du

a



a+η1 (b,a)

(ξ(u, c)ψ(u, c + η2 (d, c)) + ξ(u, c + η2 (d, c))ψ(u, c))du.

a

(6.70)

Using (6.69) and (6.70) in (6.68), we get     1 1 1 1 8ξ a + η1 (b, a), c + η2 (d, c) ψ a + η1 (b, a), c + η2 (d, c) 2 2 2 2  a+η1 (b,a)  c+η2 (d,c) 2 ξ(u, v)ψ(u, v)dvdu ⊇ η1 (b, a)η2 (d, c) a c  c+η2 (d,c) 1 (ξ(a, v)ψ(a, v) + ξ(a + η1 (b, a), v)ψ(a + η1 (b, a), v) + 3η2 (d, c) c + 2ξ(a, v)ψ(a + η1 (b, a), v) + 2ξ(a + η1 (b, a), v)ψ(a, v))dv  a+η1 (b,a) 1 (ξ(u, c)ψ(u, c) + ξ(u, c + η2 (d, c))ψ(u, c + η2 (d, c)) + 3η1 (b, a) a + 2ξ(u, c)ψ(u, c + η2 (d, c)) + 2ξ(u, c + η2 (d, c))ψ(u, c))du 1 1 2 1 + N1 (a, b, c, d) + N2 (a, b, c, d) + N3 (a, b, c, d) + N4 (a, b, c, d). (6.71) 18 9 9 9 Applying Corollary 6.3.3 for each integral in right side of (6.71), we obtain our desired result. Remark 6.4.4. If we put η1 (b, a) = b − a and η2 (d, c) = d − c in Theorem 6.4.5, we obtain Theorem 9 of [183], i.e.,

4ξ ⊇



a+b c+d , 2 2



ψ



a+b c+d , 2 2



 b d 1 ξ(u, v)ψ(u, v)dvdu (b − a)(d − c) a c 7 7 2 5 + N1∗ (a, b, c, d) + N2∗ (a, b, c, d) + N3∗ (a, b, c, d) + N4∗ (a, b, c, d), 36 36 36 9

where N1∗ (a, b, c, d), N2∗ (a, b, c, d), N3∗ (a, b, c, d), and N4∗ (a, b, c, d) are defined as previous.

Hermite–Hadamard Type Inclusions for Interval-Valued Generalized...

6.5

177

Hermite–Hadamard Type Fractional Inclusions for Harmonically h-Preinvex Interval-Valued Functions

In this section, first, we define harmonically h-preinvex interval-valued function and discuss some special cases of harmonically h-preinvex intervalvalued function [91]. Definition 6.5.1. Let h : [0, 1] ⊆ J → R be a nonnegative function such that h ≡ 0, and K ⊆ R\{0} be a harmonic invex set with respect to η(., .). Let ξ : K ⊆ R\{0} → R+ I be an interval-valued function on set K, then ξ is called harmonically h-preinvex interval-valued function with respect to η(., .) if   x(x + η(y, x)) ξ ⊇ h(1−δ)ξ(x)+h(δ)ξ(y), ∀ δ ∈ [0, 1] and ∀ x, y ∈ K. x + (1 − δ)η(y, x)

Now, we consider some special cases of harmonically h-preinvex intervalvalued functions. (1) For h(δ) = 1, function ξ is called a harmonically P −preinvex intervalvalued function. (2) For h(δ) = δ, function ξ is called a harmonically preinvex interval-valued function. (3) If h(δ) = δ s , s ∈ (0, 1), then we get the definition of Breckner type of s−harmonically preinvex interval-valued functions.

(4) If h(δ) = δ −s , s ∈ (0, 1), then we get the definition of Godunova-Levin type of s−harmonically preinvex interval-valued functions.   1 Example 6.5.1. Let K = [1, 2] ⊂ R\{0}, ξ(x) = 1 − 2x1 2 , 1 + 2x , η(y, x) = y − 2x, h(δ) = δ then ξ is harmonically h-preinvex interval-valued function on K. Now, we establish fractional inclusion of Hermite–Hadamard for harmonically h-preinvex interval-valued functions [91]. Theorem 6.5.1. Let h : [0, 1] → R be a nonnegative function such that h( 12 ) = 0. Let ξ : K = [c, c + η(d, c)] ⊆ R\{0} → R+ I be a harmonically h-preinvex interval-valued function such that ξ = [ξ, ξ] and c, d ∈ K with c < c + η(d, c). If ξ ∈ L[c, c + η(d, c)], α > 0 and η holds Condition C, then    α    1 2c(c + η(d, c)) c(c + η(d, c)) 1 α   ⊇ Γ(α) J + (ξoΩ) 1 ξ 1 2c + η(d, c) η(d, c) c αh( 2 ) c+η(d,c)   1 +Jα1 − (ξoΩ) c + η(d, c) c  1 ⊇ [ξ(c) + ξ(c + η(d, c))] δ α−1 [h(δ) + h(1 − δ)]dδ, 0

178

Integral Inequalities and Generalized Convexity

where Ω(x) =

1 x

and ξoΩ is defined by ξoΩ(x) = ξ(Ω(x)), ∀ x ∈



1 1 c+η(d,c) , c



.

Proof Since ξ is harmonically h-preinvex interval-valued function on [c, c + η(d, c)], we have   2x(x + η(y, x)) 1 ξ ⊇ ξ(x) + ξ(y), ∀ x, y ∈ [c, c + η(d, c))]. (6.72) 2x + η(y, x) h( 12 ) Let x = we get

c(c+η(d,c)) c+(1−δ)η(d,c)

1 ξ h( 12 )



and y =

2c(c + η(d, c)) 2c + η(d, c)



c(c+η(d,c)) c+δη(d,c) .

⊇ξ



Then, using Condition C in (6.72),

c(c + η(d, c)) c + (1 − δ)η(d, c)



+ξ



 c(c + η(d, c)) . c + δη(d, c) (6.73)

Multiplying (6.73) by δ α−1 , α > 0 and integrating over [0, 1] with respect to δ, we have    1 1 2c(c + η(d, c)) α−1 (IR) dδ δ ξ 2c + η(d, c) h( 12 ) 0    1 c(c + η(d, c)) α−1 dδ δ ξ ⊇ (IR) c + (1 − δ)η(d, c) 0    1 c(c + η(d, c)) + (IR) dδ. (6.74) δ α−1 ξ c + δη(d, c) 0 Applying Theorem 6.2.1 in above relation, we get    1 2c(c + η(d, c)) α−1 dδ δ ξ (IR) 2c + η(d, c) 0     1 2c(c + η(d, c)) α−1 = (R) dδ, δ ξ 2c + η(d, c) 0     1 2c(c + η(d, c)) (R) dδ δ α−1 ξ 2c + η(d, c) 0      1 1 2c(c + η(d, c)) 2c(c + η(d, c)) ξ , ξ = α 2c + η(d, c) α 2c + η(d, c)   1 2c(c + η(d, c)) = ξ , α 2c + η(d, c)  c(c + η(d, c)) dδ δ ξ (IR) c + (1 − δ)η(d, c) 0     1 c(c + η(d, c)) = (R) dδ, δ α−1 ξ c + (1 − δ)η(d, c) 0     1 c(c + η(d, c)) α−1 (R) dδ δ ξ c + (1 − δ)η(d, c) 0 

1

α−1

(6.75)



(6.76)

Hermite–Hadamard Type Inclusions for Interval-Valued Generalized... 179  α      c(c + η(d, c)) 1 1 = Γ(α) , Jα 1 + ξoΩ Jα 1 + ξoΩ η(d, c) c c c+η(d,c) c+η(d,c)  α   c(c + η(d, c)) 1 = Γ(α) . (6.77) Jα 1 + (ξoΩ) η(d, c) c c+η(d,c)

Similarly,

 c(c + η(d, c)) dδ c + δη(d, c) 0  α   c(c + η(d, c)) 1 = Γ(α) . Jα1 − (ξoΩ) η(d, c) c + η(d, c) c

(IR)



1

tα−1 ξ



Using (6.75), (6.77) and (6.78) in (6.74), we have   1 2c(c + η(d, c)) ξ 2c + η(d, c) αh( 12 ) α     1 c(c + η(d, c)) Jα 1 + (ξoΩ) ⊇ Γ(α) η(d, c) c c+η(d,c)   1 +Jα1 − (ξoΩ) . c + η(d, c) c

(6.78)

(6.79)

Since, ξ is an harmonically h-preinvex interval-valued function on [c, c + η(d, c)], we have     c(c + η(d, c)) (c + η(d, c))(c + η(d, c) + η(c, c + η(d, c)) ξ =ξ c + (1 − δ)η(d, c) c + η(d, c) + δη(c, c + η(d, c)) ⊇ h(δ)ξ(c + η(d, c)) + h(1 − δ)ξ(c)

(6.80)

and ξ



c(c + η(d, c)) c + δη(d, c)



=ξ



(c + η(d, c))(c + η(d, c)) + η(c, c + η(d, c)) c + η(d, c)) + (1 − δ)η(c, c + η(d, c))

⊇ h(1 − δ)ξ(c + η(d, c)) + h(δ)ξ(c).

 (6.81)

Adding (6.80) and (6.81), we have     c(c + η(d, c)) c(c + η(d, c)) ξ +ξ c + (1 − δ)η(d, c) c + δη(d, c) ⊇ [h(δ) + h(1 − δ)][ξ(c) + ξ(c + η(d, c))].

(6.82)

180

Integral Inequalities and Generalized Convexity

Multiplying (6.82) by δ α−1 and integrating over [0, 1] with respect to δ, we have    1 c(c + η(d, c)) α−1 (IR) dδ δ ξ c + (1 − δ)η(d, c) 0    1 c(c + η(d, c)) + (IR) dδ δ α−1 ξ c + δη(d, c) 0  1 ⊇ (IR) δ α−1 [h(δ) + h(1 − δ)][ξ(c) + ξ(c + η(d, c))]dδ. 0

This implies α       1 1 c(c + η(d, c)) α α + J 1 − (ξoΩ) J 1 + (ξoΩ) Γ(α) η(d, c) c c + η(d, c) c c+η(d,c)  1 ⊇ [ξ(c) + ξ(c + η(d, c))] δ α−1 [h(δ) + h(1 − δ)]dδ. (6.83) 0

From (6.79) and (6.83), we get   1 2c(c + η(d, c)) ξ 2c + η(d, c) αh( 12 ) α     1 c(c + η(d, c)) α   J ⊇ Γ(α) + (ξoΩ) 1 η(d, c) c c+η(d,c)   1 +Jα1 − (ξoΩ) c + η(d, c) c  1 ⊇ [ξ(c) + ξ(c + η(d, c))] δ α−1 [h(δ) + h(1 − δ)]dδ. 0

Example 6.5.2. Let K = [c, c + η(d, c))] = [1, 2], η(d, c) = d − 2c. Let α = 1 and h(δ) = δ ∀ δ ∈ [0, 1], ξ : K → R+ I be defined by   1 1 ξ(x) = − + 2, + 2 , ∀ x ∈ K. x x Then all assumptions of Theorem 6.5.1 are satisfied. Now we present some particular cases of Theorem 6.5.1. Corollary 6.5.1. If α = 1, then Theorem 6.5.1 gives the following result: 1 ξ h( 12 )



2c(c + η(d, c)) 2c + η(d, c)



⊇

2c(c + η(d, c)) η(d, c)



c+η(d,c) c

⊇ [ξ(c) + ξ(c + η(d, c))]



0

ξ(x) dx x2

1

[h(δ) + h(1 − δ)]dδ.

Hermite–Hadamard Type Inclusions for Interval-Valued Generalized...

181

Corollary 6.5.2. If h(δ) = δ, then Theorem 6.5.1 gives the following result:   2c(c + η(d, c)) ξ 2c + η(d, c))  α      1 1 Γ(α + 1) c(c + η(d, c)) α α ⊇ + J  1 − ξ J 1 + ξ 2 η(d, c) c c + η(d, c) c c+η(d,c) ⊇

ξ(c) + ξ(c + η(d, c)) . 2

Remark 6.5.1. If we put η(d, c) = d−c in above theorem, we obtain Theorem 5 of [162], i.e.   1 2cd ξ c+d αh( 12 ) α       1 1 cd Jα1 + (ξoΩ) + Jα1 − (ξoΩ) ⊇ Γ(α) d−c c d d c  1 ⊇ [ξ(c) + ξ(d)] δ α−1 [h(δ) + h(1 − δ)]dδ. 0

Remark 6.5.2. If we put η(d, c) = d − c and α = 1 in above theorem, we obtain Theorem 1 of [185], i.e.,   2cd 1 ξ c+d h( 12 ) α       1 1 cd + Jα1 − (ξoΩ) Jα1 + (ξoΩ) ⊇ d−c c d d c  1 ⊇ [ξ(c) + ξ(d)] [h(δ) + h(1 − δ)]dδ. 0

Remark 6.5.3. If we put η(d, c) = d − c and h(δ) = δ in above theorem, we obtain Theorem 3.6 of [99], i.e.,   2cd 2 ξ α c+d  α      cd 1 1 ⊇ Γ(α) + Jα1 − (ξoΩ) Jα1 + (ξoΩ) d−c c d d c 1 ⊇ [ξ(c) + ξ(d)]. α Next, we prove fractional inclusions of Hermite–Hadamard-type for the product of two harmonically h-preinvex interval-valued functions [91]. Theorem 6.5.2. Let h1 , h2 : [0, 1] → R be nonnegative functions and h1 , h2 ≡ 0 Let ξ, ψ : K = [c, c+η(d, c)] ⊆ R\{0} → R+ I be two harmonically h1 - and h2 preinvex interval-valued functions, respectively, such that ξ = [ξ, ξ], ψ = [ψ, ψ]

182

Integral Inequalities and Generalized Convexity

and c, d ∈ K with c < c + η(d, c)). If ξψ ∈ L1 [c, c + η(d, c)], α > 0 and η holds Condition C, then 

α      c(c + η(d, c)) 1 1 α Γ(α) (ψoΩ) J 1 + (ξoΩ) η(d, c) c c c+η(d,c)     1 1 +Jα1 − (ξoΩ) (ψoΩ) c + η(d, c) c + η(d, c) c ⊇ F (c, c + η(d, c))



1

0

+ G(c, c + η(d, c))

[δ α−1 + (1 − δ)α−1 ]h1 (δ)h2 (δ)dδ



1 0

[δ α−1 + (1 − δ)α−1 ]h1 (1 − δ)h2 (δ)]dδ,

(6.84)

where F (c, c + η(d, c)) = ξ(c)ψ(c) + ξ(c + η(d, c))ψ(c + η(d, c)), G(c, c + η(d, c)) = ξ(c)ψ(c + η(d, c)) + ξ(c + η(d, c))ψ(c) and Ω(x) = x1 . Proof Since ξ and ψ are two harmonically h1 - and h2 -preinvex interval-valued functions on [c, c + η(d, c))], respectively. Therefore,     (c + η(d, c))(c + η(d, c) + η(c, c + η(d, c)) c(c + η(d, c)) =ξ ξ c + (1 − δ)η(d, c) c + η(d, c) + δη(c, c + η(d, c)) ⊇ h1 (δ)ξ(c + η(d, c)) + h1 (1 − δ)ξ(c)

and ψ



c(c + η(d, c)) c + (1 − δ)η(d, c)



=ψ



(6.85)

(c + η(d, c))(c + η(d, c) + η(c, c + η(d, c)) c + η(d, c) + δη(c, c + η(d, c))

⊇ h2 (δ)ψ(c + η(d, c)) + h2 (1 − δ)ψ(c).



(6.86)

Since, ξ(x), ψ(x) ∈ R+ I , ∀ x ∈ [c, c + η(d, c)], then from (6.85) and (6.86), we obtain     c(c + η(d, c)) c(c + η(d, c)) ψ ξ c + (1 − δ)η(d, c) c + (1 − δ)η(d, c) ⊇ h1 (δ)h2 (δ)ξ(c + η(d, c))ψ(c + η(d, c)) + h1 (1 − δ)h2 (1 − δ)ξ(c)ψ(c)

+ h1 (δ)h2 (1 − δ)ξ(c + η(d, c))ψ(c) + h1 (1 − δ)h2 (δ)ξ(c)ψ(c + η(d, c)). (6.87)

Similarly,     c(c + η(d, c)) c(c + η(d, c)) ξ ψ c + δη(d, c) c + δη(d, c)

⊇ h1 (1 − δ)h2 (1 − δ)ξ(c + η(d, c))ψ(c + η(d, c)) + h1 (δ)h2 (δ)ξ(c)ψ(c)

+ h1 (1 − δ)h2 (δ)ξ(c + η(d, c))ψ(c) + h1 (δ)h2 (1 − δ)ξ(c)ψ(c + η(d, c)). (6.88)

Hermite–Hadamard Type Inclusions for Interval-Valued Generalized...

183

Adding (6.87) and (6.88), we have     c(c + η(d, c)) c(c + η(d, c)) ξ ψ c + (1 − δ)η(d, c) c + (1 − δ)η(d, c)     c(c + η(d, c)) c(c + η(d, c)) ψ +ξ c + δη(d, c) c + δη(d, c) ⊇ [h1 (δ)h2 (δ) + h1 (1 − δ)h2 (1 − δ)][ξ(c)ψ(c) + ξ(c + η(d, c))ψ(c + η(d, c))]

+ [h1 (δ)h2 (1 − δ) + h1 (1 − δ)h2 (δ)][ξ(c + η(d, c))ψ(c) + ξ(c)ψ(c + (d, r))]

= F (c, c + η(d, c))[h1 (δ)h2 (δ) + h1 (1 − δ)h2 (1 − δ)]

+ G(c, c + η(d, c))[h1 (1 − δ)h2 (δ) + h1 (δ)h2 (1 − δ)].

(6.89)

Multiplying (6.89) by δ α−1 and integrating over [0, 1] with respect to δ, we have      1 c(c + η(d, c)) c(c + η(d, c)) ψ dδ (IR) δ α−1 ξ c + (1 − δ)η(d, c) c + (1 − δ)η(d, c) 0      1 c(c + η(d, c)) c(c + η(d, c)) + (IR) ψ dδ δ α−1 ξ c + δη(d, c) c + δη(d, c) 0  1 ⊇ (IR) δ α−1 F (c, c + η(d, c))[h1 (δ)h2 (δ) + h1 (1 − δ)h2 (1 − δ)]dδ 0

+ (IR)



1

0

δ α−1 G(c, c + η(d, c))[h1 (1 − δ)h2 (δ) + h1 (δ)h2 (1 − δ)]dδ. (6.90)

Since,    c(c + η(d, c)) c(c + η(d, c)) (IR) ψ dδ δ ξ c + (1 − δ)η(d, c) c + (1 − δ)η(d, c) 0  α     c(c + η(d, c)) 1 1 α = Γ(α) (ψoΩ) , (6.91) J 1 + (ξoΩ) η(d, c) c c c+η(d,c) 

1

α−1



   c(c + η(d, c)) c(c + η(d, c)) (IR) ψ dδ δ ξ c + δη(d, c) c + δη(d, c) 0  α    c(c + η(d, c)) 1 α   = Γ(α) J 1 − (ξoΩ) η(d, c) c + η(d, c) c   1 ×(ψoΩ) . c + η(d, c) 

1

α−1



(6.92)

184

Integral Inequalities and Generalized Convexity

Using (6.91), (6.92) in (6.90), we have α       1 1 c(c + η(d, c)) (ψoΩ) Jα 1 + (ξoΩ) Γ(α) η(d, c) c c c+η(d,c)     1 1 +Jα1 − (ξoΩ) (ψoΩ) c + η(d, c) c + η(d, c) c  1 ⊇ F (c, c + η(d, c)) [δ α−1 + (1 − δ)α−1 ]h1 (δ)h2 (δ)dδ 0

+ G(c, c + η(d, c))



1

0

[δ α−1 + (1 − δ)α−1 ]h1 (1 − δ)h2 (δ)]dδ.

Corollary 6.5.3. If α = 1, then Theorem 6.5.2 gives the following result:  c(c + η(d, c)) c+η(d,c)) ξ(x)ψ(x) dx η(d, c) x2 c  1  1 h1 (δ)h2 (δ)dδ + G(c, c + η(d, c)) h1 (1 − δ)h2 (δ)]dδ. ⊇ F (c, c + η(d, c)) 0

0

Corollary 6.5.4. If h1 (δ) = h2 (δ) = δ, then Theorem 6.5.2 gives the following result: α      1 1 c(c + η(d, c)) α   J (ψoΩ) Γ(α) + (ξoΩ) 1 η(d, c) c c c+η(d,c)     1 1 +Jα1 − (ξoΩ) (ψoΩ) c + η(d, c) c + η(d, c) c  1 ⊇ F (c, c + η(d, c)) δ 2 [δ α−1 + (1 − δ)α−1 ]dδ 

0

+ G(c, c + η(d, c)) =



1

δ(1 − δ)[δ α−1 + (1 − δ)α−1 ]dδ

0

2 α2 + α + 2 F (c, c + η(d, c)) + G(c, c + η(d, c)). α(α + 1)(α + 2) (α + 1)(α + 2)

Remark 6.5.4. If we put η(d, c) = d−c in Theorem 6.5.2, we obtain Theorem 6 of [162], i.e., α       1 cd 1 (ψoΩ) Γ(α) Jα1 + (ξoΩ) d − c) c c d     1 1 +Jα1 − (ξoΩ) (ψoΩ) d d c  1 ⊇ F (c, d) [δ α−1 + (1 − δ)α−1 ]h1 (δ)h2 (δ)dδ 0

+ G(c, d)



1

0

[δ α−1 + (1 − δ)α−1 ]h1 (1 − δ)h2 (δ)]dδ.

Hermite–Hadamard Type Inclusions for Interval-Valued Generalized...

185

Remark 6.5.5. If we put η(d, c) = d − c and α = 1 in Theorem 6.5.2, we obtain Theorem 3 of [185], i.e.,  α      cd 1 1 (ψoΩ) Jα1 + (ξoΩ) d − c) c c d     1 1 +Jα1 − (ξoΩ) (ψoΩ) d d c  1  1 ⊇ 2F (c, d) h1 (δ)h2 (δ)dδ + 2G(c, d) h1 (1 − δ)h2 (δ)]dδ. 0

0

Theorem 6.5.3. Let h1 , h2 : [0, 1] → R be nonnegative functions and h1 ( 12 )h2 ( 12 ) = 0. Let ξ, ψ : K = [c, c + η(d, c)] ⊆ R\{0} → R+ I be two harmonically h1 - and h2 -preinvex interval-valued functions, respectively, such that ξ = [ξ, ξ], ψ = [ψ, ψ] and c, d ∈ K with c < c+η(d, c)). If ξψ ∈ L[c, c+η(d, c)], α > 0 and η holds Condition C, then     2c(c + η(d, c)) 2c(c + η(d, c)) 1 ξ ψ 2c + η(d, c) 2c + η(d, c) αh1 ( 12 )h2 ( 12 ) α       1 1 c(c + η(d, c)) (ψoΩ) Jα 1 + (ξoΩ) ⊇ Γ(α) η(d, c) c c c+η(d,c)     1 1 +Jα1 − (ξoΩ) (ψoΩ) c + η(d, c) c + η(d, c) c  1 + F (c, c + η(d, c)) (δ α−1 + (1 − δ)α−1 )h1 (δ)h2 (1 − δ)dδ 0

+ G(c, c + η(d, c))



1

0

(δ α−1 + (1 − δ)α−1 )h1 (δ)h2 (δ)dδ,

where F (c, c + η(d, c)) and G(c, c + η(d, c)) are defined as previous. Proof Since ξ is harmonically h1 -preinvex interval-valued function on [c, c + η(d, c))], we have   2x(x + η(y, x)) 1 ξ ⊇ ξ(x) + ξ(y), ∀ x, y ∈ [c, c + η(d, c)]. (6.93) 2x + η(y, x) h1 ( 12 ) Let u = we get

c(c+η(d,c)) c+(1−δ)η(d,c)

1 ξ h1 ( 12 )



and v =

2c(c + η(d, c)) 2c + η(d, c)



c(c+η(d,c)) c+δη(d,c) .

⊇ξ



Then, using Condition C in (6.93),

c(c + η(d, c)) c + (1 − δ)η(d, c)



+ξ



 c(c + η(d, c)) . c + δη(d, c) (6.94)

186

Integral Inequalities and Generalized Convexity

Similarly, 1 ψ h2 ( 12 )



2c(c + η(d, c)) 2c + η(d, c)



⊇ψ



c(c + η(d, c)) c + (1 − δ)η(d, c)



+ψ



 c(c + η(d, c)) . c + δη(d, c) (6.95)

From (6.94) and (6.95), we get     2c(c + η(d, c)) 2c(c + η(d, c)) 1 ξ ψ 2c + η(d, c) 2c + η(d, c) h1 ( 12 )h2 ( 12 )      c(c + η(d, c)) c(c + η(d, c)) +ξ ⊇ ξ c + (1 − δ)η(d, c) c + δη(d, c)      c(c + η(d, c)) c(c + η(d, c)) +ψ × ψ c + (1 − δ)η(d, c) c + δη(d, c) 

   c(c + η(d, c)) c(c + η(d, c)) =ξ ψ c + (1 − δ)η(d, c) c + (1 − δ)η(d, c)     c(c + η(d, c)) c(c + η(d, c)) ψ +ξ c + δη(d, c) c + δη(d, c)      c(c + η(d, c)) c(c + η(d, c)) + ξ ψ c + (1 − δ)η(d, c) c + δη(d, c)     c(c + η(d, c)) c(c + η(d, c)) ψ . +ξ c + δη(d, c) c + (1 − δ)η(d, c)

(6.96)

Since ξ(x) and ψ(x) ∈ X+ I , ∀x ∈ [c, c + η(d, c)] are two harmonically h1 - and h2 - preinvex interval-valued functions, respectively. Therefore,     c(c + η(d, c)) c(c + η(d, c)) ξ ψ c + (1 − δ)η(d, c) c + δη(d, c) ⊇ h1 (δ)h2 (δ)ξ(c + η(d, c))ψ(c) + h1 (1 − δ)h2 (1 − δ)ξ(c)ψ(c + η(d, c))

+ h1 (δ)h2 (1 − δ)ξ(c + η(d, c))ψ(c + η(d, c)) + h1 (1 − δ)h2 (δ)ξ(c)ψ(c). (6.97)

Similarly,     c(c + η(d, c)) c(c + η(d, c)) ξ ψ c + δη(d, c) c + (1 − δ)η(d, c)

⊇ h1 (δ)h2 (δ)ξ(c))ψ(c + η(d, c)) + h1 (1 − δ)h2 (1 − δ)ξ(c + η(d, c))ψ(c)

+ h1 (δ)h2 (1 − δ)ξ(c)ψ(c) + h1 (1 − δ)h2 (δ)ξ(c + η(d, c))ψ(c + η(d, c)). (6.98)

Hermite–Hadamard Type Inclusions for Interval-Valued Generalized...

187

Adding (6.97) and (6.98), we obtain     c(c + η(d, c)) c(c + η(d, c)) ξ ψ c + (1 − δ)η(d, c) c + δη(d, c)     c(c + η(d, c)) c(c + η(d, c)) +ξ ψ c + δη(d, c) c + (1 − δ)η(d, c) ⊇ G(c, c + η(d, c))[h1 (δ)h2 (δ) + h1 (1 − δ)h2 (1 − δ)] + F (c, c + η(d, c))[h1 (1 − δ)h2 (δ) + h1 (δ)h2 (1 − δ)].

(6.99)

From (6.96) and (6.99), we have     2c(c + η(d, c)) 1 2c(c + η(d, c)) ξ ψ 2c + η(d, c) 2c + η(d, c) h1 ( 12 )h2 ( 12 )     c(c + η(d, c)) c(c + η(d, c)) ψ ⊇ξ c + (1 − δ)η(d, c) c + (1 − δ)η(d, c)     c(c + η(d, c)) c(c + η(d, c)) +ξ ψ c + δη(d, c) c + δη(d, c) + G(c, c + η(d, c))[h1 (δ)h2 (δ) + h1 (1 − δ)h2 (1 − δ)]dδ + F (c, c + η(d, c))[h1 (1 − δ)h2 (δ) + h1 (δ)h2 (1 − δ)]dδ.

(6.100)

Multiplying (6.100) by δ α−1 , then integrating over [0, 1] with respect to δ, we get      1 2c(c + η(d, c)) 2c(c + η(d, c)) 1 α−1 (IR) ψ dδ δ ξ 2c + η(d, c) 2c + η(d, c) h1 ( 12 )h2 ( 12 ) 0      1 c(c + η(d, c)) c(c + η(d, c)) ψ dδ δ α−1 ξ ⊇ (IR) c + (1 − δ)η(d, c) c + (1 − δ)η(d, c) 0      1 c(c + η(d, c)) c(c + η(d, c)) + (IR) ψ dδ δ α−1 ξ c + δη(d, c) c + δη(d, c) 0  1 + G(c, c + η(d, c))(IR) δ α−1 [h1 (δ)h2 (δ) + h1 (1 − δ)h2 (1 − δ)]dδ 0

+ F (c, c + η(d, c))(IR)



1 0

δ α−1 [h1 (1 − δ)h2 (δ) + h1 (δ)h2 (1 − δ)]dδ.

188

Integral Inequalities and Generalized Convexity

This implies,     2c(c + η(d, c)) 2c(c + η(d, c)) 1 ξ ψ 2c + η(d, c) 2c + η(d, c) αh1 ( 12 )h2 ( 12 ) α       1 1 c(c + η(d, c)) ψoΩ Jα 1 + (ξoΩ) ⊇ Γ(α) η(d, c) c c c+η(d,c)     1 1 +Jα1 − (ξoΩ) (ψoΩ) c + η(d, c)) c + η(d, c)) c  1 + F (c, c + η(d, c)) (δ α−1 + (1 − δ)α−1 )h1 (δ)h2 (1 − δ)]dδ 0

+ G(c, c + η(d, c))



1

(δ α−1 + (1 − δ)α−1 )h1 (δ)h2 (δ)dδ.

0

Corollary 6.5.5. If α = 1, then Theorem 6.5.3 gives the following result:     2c(c + η(d, c)) 1 2c(c + η(d, c)) ξ ψ 2c + η(d, c) 2c + η(d, c) 2h1 ( 12 )h2 ( 12 )  c+η(d,c)) c(c + η(d, c)) ξ(x)ψ(x) dx ⊇ η(d, c) x2 c  1 h1 (δ)h2 (1 − δ)dδ + F (c, c + η(d, c)) 0

+ G(c, c + η(d, c))



1

h1 (δ)h2 (δ)dδ. 0

Corollary 6.5.6. If h1 (δ) = h2 (δ) = δ, then Theorem 6.5.3 gives the following result:    2c(c + η(d, c)) 2c(c + η(d, c)) ψ 2c + η(d, c) 2c + η(d, c)  α      c(c + η(d, c)) 1 1 ⊇ Γ(α)ξ Jα 1 + (ξoΩ) ψoΩ η(d, c) c c c+η(d,c)     1 1 +Jα1 − (ξoΩ) (ψoΩ) c + η(d, c) c + η(d, c) c  1 + F (c, c + η(d, c)) δ(1 − δ)(δ α−1 + (1 − δ)α−1 )dδ

4 ξ α



0

+ G(c, c + η(d, c))



1

0

δ 2 (δ α−1 + (1 − δ)α−1 )dδ

Hermite–Hadamard Type Inclusions for Interval-Valued Generalized...  α      c(c + η(d, c)) 1 1 = Γ(α)ξ ψoΩ Jα 1 + (ξoΩ) η(d, c) c c c+η(d,c)     1 1 +Jα1 − (ξoΩ) (ψoΩ) c + η(d, c) c + η(d, c) c +

189

α2 + α + 2 2 F (c, c + η(d, c)) + G(c, c + η(d, c)). (α + 1)(α + 2) α(α + 1)(α + 2)

Remark 6.5.6. If we put η(d, c) = d−c in above theorem, we obtain Theorem 7 of [162], i.e.,     2cd 2cd 1 ξ ψ c+d c+d αh1 ( 12 )h2 ( 12 ) α       1 1 cd (ψoΩ) Jα1 + (ξoΩ) ⊇ Γ(α) d−c c c d     1 1 +Jα1 − (ξoΩ) (ψoΩ) d d c  1 + F (c, d) (δ α−1 + (1 − δ)α−1 )h1 (δ)h2 (1 − δ)dδ 0

+ G(c, d)



1

0

(δ α−1 + (1 − δ)α−1 )h1 (δ)h2 (δ)dδ.

Remark 6.5.7. If we put η(d, c) = d − c and α = 1 in above theorem, we obtain Theorem 4 of [185], i.e., 1 ξ h1 ( 12 )h2 ( 12 )



2cd c+d



ψ



2cd c+d



    1 1 α   J 1 + (ξoΩ) (ψoΩ) ⊇ c c d     1 1 +Jα1 − (ξoΩ) (ψoΩ) d d c  1  + 2F (c, d) h1 (δ)h2 (1 − δ)dδ + 2G(c, d) 

cd d−c



0

1

h1 (δ)h2 (δ)dδ. 0

Chapter 7 Some Inequalities for Multidimensional General h-Harmonic Preinvex and Strongly Generalized Convex Stochastic Processes

7.1

Introduction

Dragomir [45] established Hermite–Hadamard inequality for convex functions on the coordinates on a rectangle from the plane R2 . It is well known that every convex mapping ξ : [a, b] × [c, d] → R is convex on the coordinates, but the converse is not true in general. Noor et al. [126] introduced harmonic preinvex function and obtained several new Hermite–Hadamard type inequalities for harmonic preinvex functions. Further, Noor et al. [121] defined the relative harmonic preinvex functions and derived some integral inequalities Hermite–Hadamard, Simpson’s and trapezoidal for the relative harmonic preinvex function. If T is some index set and S is the common sample space of the random variable, then the collection of random variables {ξt (s), t ∈ T, s ∈ S} is called stochastic process. Allen [3] introduced basic theory of stochastic processes and applied these methods to biological problems, such as enzyme kinetics, population extinction, the spread of epidemics and the genetics of inbreeding. Nikodem [116] introduced some powerful properties of convex stochastic processes. He proved that a convex stochastic processes ξ : K × Ω → R is continuous if and only if for all x, y ∈ K and δ ∈ [0, 1] ξ(δx + (1 − δ)y, .) ≤ δξ(x, .) + (1 − δ)ξ(y, .) almost everywhere, where R denotes the extended real line. Kuhn [89] studied the convex stochastic programs with a generalized non convex dependence on the random parameters. Kuhn [89] proved that, under certain conditions, the saddle structure can be restored by adding specific random variables to the profit functions. These random variables are referred to as ‘correction terms’.

DOI: 10.1201/9781003408284-7

191

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Integral Inequalities and Generalized Convexity

Kotrys [88] defined the strongly convex stochastic processes and derived the Hermite–Hadamard inequality, Jensen inequality and Kuhn and Bernstein theorem. A stochastic process ξ : K × Ω → R is strongly δ−convex with modulus µ(.) if and only if the stochastic process φ : K × Ω → R defined by φ(x, .) = ξ(x, .) − µ(.)x2 is δ−convex [88]. Further, Karahan and Okur [71] obtained some generalized Hermite–Hadamard inequalities for convex stochastic process on the coordinates. Further, Okur and Aliyev [129] introduced general preinvex and multidimensional general preinvex stochastic processes, and derived Hermite–Hadamard inequality for these processes. Further, Ibrahim [63] introduced the concept of strongly h−convex stochastic process. Jung et al. [69] introduced the notion of η-convex stochastic processes and established Jensen, Hermite–Hadamard and Ostrowski type inequalities. ˙ scan and Improved power mean inteFurther, Fu et al. [51] derived H¨ older-I¸ gral inequalities, and proved Hermite–Hadamard type integral inequalities for n−polynomial stochastic processes. The organization of the chapter is as follows: In Section 7.2, we recall some basic results that are necessary for our main results. In Section 7.3, we introduce the concept of general h−harmonic preinvexity for real-valued stochastic processes (Gh−HPηϕ SP ) and discuss some special cases of our definition. We prove Hermite–Hadamard type inequalities for Gh−HPηϕ SP . Further, in Section 7.4, we define multidimensional general h−harmonic preinvex stochastic process and special cases in favor of the definition. Also, we obtain Hermite– Hadamard type inequalities for multidimensional general h−harmonic preinvex stochastic processes. In Section 7.5, we introduce the concept of strongly η-convex stochastic processes. Further, we prove the Hermite–Hadamard inequality, Ostrowski inequality and some other interesting inequalities.

7.2

Preliminaries

In this section, we mention some definitions and related results required for this chapter. We suppose that K be a nonempty closed set. Also suppose that η(., .) : K × K → R and ϕ : K → K be arbitrary functions. Noor et al. [121] derived the following Hermite–Hadamard inequality for relative harmonic preinvex functions. Theorem 7.2.1. Let ξ : K = [c, c+η(d, c)] ⊆ R\{0} → R be relative harmonic preinvex function with c < c + η(d, c). If ξ ∈ L[c, c + η(d, c)] and Condition C holds, then 1 ξ 2h( 12 )



2c(c + η(d, c)) 2c + η(d, c)



c(c + η(d, c)) ≤ η(d, c)  ≤ [ξ(c) + ξ(d)]



0

c+η(d,c)

c 1

h(δ)dδ.

ξ(x) dx x2 (7.1)

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 193 Okur and Aliyev [129] introduced the concepts of general invex set and preinvex functions. Definition 7.2.1. A set K is said to be general invex set with respect to η(., .) and ϕ, if ϕ(x) + δη(ϕ(y), ϕ(x)) ∈ K, ∀ x, y ∈ R : ϕ(x), ϕ(y) ∈ K, δ ∈ [0, 1]. Definition 7.2.2. A function ξ on K is said to be general preinvex with respect to arbitrary function η and φ, if ξ(ϕ(x) + δη(ϕ(y), ϕ(x))) ≤ (1 − δ)ξ(ϕ(x)) + δξ(ϕ(y)), ∀ x, y ∈ R :

ϕ(x), ϕ(y) ∈ K, δ ∈ [0, 1].

(7.2)

Okur and Aliyev [129] gave the following Condition C for GPηϕ SP. Definition 7.2.3. Let η : K × K → R hold the following criterions: η(ϕ(x), ϕ(x) + δη(ϕ(y)ϕ(x))) = −δη(ϕ(y), ϕ(x)); η(ϕ(y), ϕ(x) + δη(ϕ(y)ϕ(x))) = (1 − δ)η(ϕ(y), ϕ(x));

∀ x, y ∈ R : ϕ(x), ϕ(y) ∈ K, δ ∈ [0, 1].

The following Hermite–Hadamard inequality for general preinvex functions is obtained by Awan et al. [10]. Theorem 7.2.2. Let ξ : K = [ϕ(c), ϕ(c) + η(ϕ(d), ϕ(c))] → R be a general preinvex function with η(ϕ(d), ϕ(c)) > 0. If η(., .) satisfies the Condition C, then we have    ϕ(c)+η(ϕ(d),ϕ(c)) 1 2ϕ(c) + η(ϕ(d), ϕ(c)) ≤ ξ(ϕ(x))dϕ(x) ξ 2 η(ϕ(d), ϕ(c)) ϕ(c) ≤

ξ(ϕ(c)) + ξ(ϕ(d)) . 2

(7.3)

Okur and Aliyev [129] defined multidimensional general preinvex stochastic processes. Definition 7.2.4. A stochastic process ξ : Kηϕ × Ω → R is said to be multidimensional general preinvex (M GPηϕ SP ) with respect to η and ϕ on Kηϕ if the following inequality holds almost everywhere ξ((ϕ(y) + δη(ϕ(x), ϕ(y))), .) ≤ δξ(ϕ(x), .) + (1 − δ)ξ(ϕ(y), .), for all ϕ(x), ϕ(y) ∈ Kηϕ and δ ∈ [0, 1], where Kηϕ ⊆ Rn be a non empty closed set, and η : Kηϕ × Kηϕ → Rn and ϕ : Kηϕ → Kηϕ be arbitrary functions. Kotrys [87] proved the Hermite–Hadamard inequality for convex stochastic processes.

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Integral Inequalities and Generalized Convexity

Theorem 7.2.3. Let ξ : I × Ω → R is convex and mean square continuous in the interval I. Then for any c, d ∈ I, we have    d c+d 1 ξ(c, .) + ξ(d, .) ξ ,. ≤ (a.e.). ξ(u, .)du ≤ 2 d−c c 2 The following definition of strongly convex stochastic processes is given by Kotrys [88]. Definition 7.2.5. Let µ : Ω → R denote a positive random variable. The stochastic process ξ : I×Ω → R is called strongly convex with modulus µ(.) > 0, if for all δ ∈ [0, 1] and x, y ∈ I the inequality ξ(δx + (1 − δ)y, .) ≤ δξ(x, .) + (1 − δ)ξ(y, .) − µ(.)δ(1 − δ)(x − y)2 (a.e.) is satisfied. Kotrys [88] proved the following Hermite–Hadamard inequality for strongly Jensen convex stochastic processes. Theorem 7.2.4. Let ξ : I × Ω → R be a stochastic process, which is strongly Jensen convex with modulus µ(.) and mean square continuous in the interval I. Then for any c, d ∈ I, we have    d c+d µ(.) 1 ξ(c, .) + ξ(d, .) 2 ξ ,. + (d − c) ≤ ξ(u, .)du ≤ 2 12 d−c c 2 µ(.) (d − c)2 (a.e.). − 6 Jung et al. [69] introduced the concept of η-convex stochastic processes. Definition 7.2.6. Let (Ω, A, P ) be a probability space and I ⊆ R be an interval, then ξ : I × Ω → R is an η-convex stochastic process if ξ(δx + (1 − δ)y, .) ≤ ξ(y, .) + δη(ξ(x, .), ξ(y, .)) (a.e.), ∀x, y ∈ I and δ ∈ [0, 1]. ˙ scan and Improved power mean Fu et al. [51] derived the following H¨ older-I¸ integral inequalities. ˙scan integral inequality) Let ξ, φ : [c, d] × Ω → R Theorem 7.2.5. (H¨ older-I¸ be real stochastic process and |ξ  |p , |φ |p be mean square integrable on [c, d]. If p > 1 and p1 + 1q = 1, then the following inequality holds almost everywhere: 

d

|ξ(x, .)φ(x, .)|dx  1/p  1/q  d d 1  p q ≤ (d − x)|ξ(x, .)| dx (d − x)|φ(x, .)| dx d−c c c  1/p  1/q  d d . + (x − c)|ξ(x, .)|p dx (x − c)|φ(x, .)|q dx c

c

c

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 195 Theorem 7.2.6. (Improved power mean integral inequality) Let ξ, φ : [c, d] × Ω → R be real stochastic process and |ξ|, |ξ||φ|q be mean square integrable on [c, d]. If q ≥ 1, then the following inequality holds almost everywhere:  1− q1  d  d 1  |ξ(x, .)φ(x, .)|dx ≤ (d − x)|ξ(x, .)|dx d−c c c × +

 

d

q

(d − x)|ξ(x, .)||φ(x, .)| dx

c d c

(x − c)|ξ(x, .)|dx

1/q

1− q1 

d c

1/q  . (x − c)|ξ(x, .)||φ(x, .)|q dx

The following lemmas are obtained by Gonzales et al. [53] and Fu et al. [51], respectively. Lemma 7.2.1. Let ξ : I ×Ω → R be a stochastic process which is mean square differentiable on I 0 . If ξ  is mean square integrable on [c, d], where c, d ∈ I with c < d, then the following equality holds  d  (x − c)2 1  1 ξ(u, .)du = tξ (tx + (1 − t)c, .)dt ξ(t, .) − d−c c d−c 0  (d − x)2 1  − tξ (tx + (1 − t)d, .)dt, (a.e), for each x ∈ [c, d]. d−c 0

Lemma 7.2.2. Let ξ : I × Ω → R be a mean square differentiable stochastic process on I 0 and ξ  is mean square integrable on [c, d], where c, d ∈ I, c < d. Then we have almost everywhere  d  1 d−c 1 ξ(c, .) + ξ(d, .) − ξ(u, .)du = (1 − 2δ)ξ  (δc + (1 − δ)d, .)dδ. 2 d−c c 2 0 Jung et al. [69] proved the Hermite–Hadamard inequality for η-convex stochastic processes.

Theorem 7.2.7. Suppose that ξ : [c, d] × Ω → R is an η-convex stochastic process such that η is bounded above on ξ[c, d] × ξ[c, d], then the following inequalities hold almost everywhere:    d Mη 1 c+d ξ ,. − ≤ ξ(u, .)du 2 2 d−c c ξ(c, .) + ξ(d, .) 1 ≤ + (η(ξ(c, .), ξ(d, .)) + η(ξ(d, .), ξ(c, .))) 2 4 ξ(c, .) + ξ(d, .) + Mη . ≤ 2 Now we are ready to discuss our main results. First, we discuss general hharmonically preinvex stochastic process and second multidimensional general h−harmonically preinvex stochastic process.

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Integral Inequalities and Generalized Convexity

7.3

General h−Harmonically Preinvex Stochastic Process (Gh − HPηϕ SP )

We give the definition of general h−harmonically preinvex stochastic process and discuss some special cases of our new definition. Further, we present Hermite–Hadamard type inequalities for these stochastic processes [155]. Definition 7.3.1. Let h : (0, 1) → R be a nonnegative function, h ≡ 0 and ξ : K × Ω ⊆ R\{0} × Ω → R be a stochastic process on the general harmonic invex set K with respect to η and ϕ. Then the stochastic process ξ is called general h−harmonic preinvex stochastic process (Gh − HPηϕ SP ) with respect to η and ϕ if and only if   ϕ(x)(ϕ(x) + η(ϕ(y), ϕ(x))) ξ , . ≤ h(1 − δ)ξ(ϕ(x), .) + h(δ)ξ(ϕ(y), .), ϕ(x) + (1 − δ)η(ϕ(y), ϕ(x)) ∀ x, y ∈ R : ϕ(x), ϕ(y) ∈ K, δ ∈ [0, 1]. (7.4)

Note that for δ = 12 , we have Jensen type Gh − HPηϕ SP with respect to η and ϕ     2ϕ(x)(ϕ(x) + η(ϕ(y)ϕ(x))) 1 ξ ,. ≤ h [ξ(ϕ(x), .) + ξ(ϕ(y), .)], 2ϕ(x) + η(ϕ(y), ϕ(x)) 2 ∀ x, y ∈ R : ϕ(x), ϕ(y) ∈ K.

(7.5)

We now discuss some special cases of Definition 7.3.1: 1 If h(δ) = δ, then Definition 7.3.1 reduces to the definition of general harmonic preinvex stochastic process (GHPηϕ SP ). 2 If h(δ) = δ s , then Definition 7.3.1 reduces to the definition of Breckner type of general s-harmonic preinvex stochastic process. 3 If h(δ) = δ −s , then Definition 7.3.1 reduces to the definition of Godunova-Levin type of general s-harmonic preinvex stochastic process. 4 If h(δ) = 1, then Definition 7.3.1 reduces to the definition of general harmonic P-preinvex stochastic process. Lemma 7.3.1. Let (Ω, ζ, P ) be an arbitrary probability space and ξ : K × Ω ⊆ (R\{0} × Ω) → R be a stochastic process on the general harmonic invex set K with respect to η and ϕ. If ϕ is a continuous function on K, then ξ can be known as 1 mean square continuous on K, 2 monotonic if it is increasing and decreasing,

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 197 3 mean-square differentiable at a point ϕ(x) ∈ K, 4 mean-square integrable on [ϕ(x), ϕ(x) + η(ϕ(y), ϕ(x))] ⊆ K. Theorem 7.3.1. Let h : (0, 1) → R. Under the assumptions of Lemma 7.3.1, let ξ : [ϕ(c), ϕ(c) + η(ϕ(d), ϕ(c))] × Ω → R be Gh − HPηϕ SP with respect to η and ϕ. If η fulfills the criterions of Condition C, then the following inequalities hold almost everywhere   1 2ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ξ ,. 2ϕ(c) + η(ϕ(d), ϕ(c)) 2h( 12 )  ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ϕ(c)+η(ϕ(d),ϕ(c)) ξ(z, .) dz ≤ η(ϕ(d), ϕ(c)) z2 ϕ(c)  1 ≤ [ξ(ϕ(c), .) + ξ(ϕ(d), .)] h(δ)dδ. 0

Proof From definition of Gh − HPηϕ SP for δ = 12 , we have     1 2ϕ(x)(ϕ(x) + η(ϕ(y), ϕ(x))) ,. ≤ h [ξ(ϕ(x), .) + ξ(ϕ(y), .)], ξ 2ϕ(x) + η(ϕ(y), ϕ(x)) 2 



∀ x, y ∈ R : ϕ(x), ϕ(y) ∈ K.

  ϕ(c)(ϕ(c)+η(ϕ(d),ϕ(c))) Applying ϕ(x) = ϕ(c)(ϕ(c)+η(ϕ(d),ϕ(c))) , ϕ(y) = and ϕ(c)+δη(ϕ(d),ϕ(c)) ϕ(c)+(1−δ)η(ϕ(d),ϕ(c)) Condition C in above inequality, we obtain   2ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ξ ,. 2ϕ(c) + η(ϕ(d), ϕ(c))     1 ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ≤h ξ ,. 2 ϕ(c) + δη(ϕ(d), ϕ(c))   ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) +ξ ,. . ϕ(c) + (1 − δ)η(ϕ(d), ϕ(c)) Integrating both sides of the above inequality with respect to δ over [0, 1], we get   2ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ξ ,. 2ϕ(c) + η(ϕ(d), ϕ(c))    1   1 ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ≤h , . dδ ξ 2 ϕ(c) + δη(ϕ(d), ϕ(c)) 0    1  ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) + , . dδ . ξ ϕ(c) + (1 − δ)η(ϕ(d), ϕ(c)) 0

198

Integral Inequalities and Generalized Convexity

This implies,   2ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) 1 ξ ,. 2ϕ(c) + η(ϕ(d), ϕ(c)) 2h( 12 ) ≤

ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) η(ϕ(d), ϕ(c))



ϕ(c)+η(ϕ(d),ϕ(c)) ϕ(c)

ξ(z, .) dz. z2

(7.6)

From the definition of Gh − HPηϕ SP , we have   ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ξ , . ≤ h(1 − δ)ξ(ϕ(c), .) + h(δ)ξ(ϕ(d), .). ϕ(c) + (1 − δ)η(ϕ(d), ϕ(c)) Integrating with respect to δ over [0, 1], we have 

ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) η(ϕ(d), ϕ(c))

≤ [ξ(ϕ(c), .) + ξ(ϕ(d), .)]

ϕ(c)+η(ϕ(d),ϕ(c)) ϕ(c)



ξ(z, .) dz z2

1

h(δ)dδ.

(7.7)

0

From (7.6) and (7.7), we get   1 2ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ξ ,. 2ϕ(c) + η(ϕ(d), ϕ(c)) 2h( 12 ) ≤

ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) η(ϕ(d), ϕ(c))

≤ [ξ(ϕ(c), .) + ξ(ϕ(d), .)]





ϕ(c)+η(ϕ(d),ϕ(c)) ϕ(c)

ξ(z, .) dz z2

1

h(δ)dδ.

0

This completes the proof. Remark 7.3.1. If h(δ) = δ and ξ is a GPηϕ SP , then above theorem reduces to Theorem 2.1 of [129], i.e.   2ϕ(c) + η(ϕ(d), ϕ(c)) ξ ,. 2 1 ≤ η(ϕ(d), ϕ(c)) ≤



ϕ(c)+η(ϕ(d),ϕ(c))

ξ(ϕ(x), .)dϕ(x) ϕ(c)

ξ(ϕ(c), .) + ξ(ϕ(d), .) . 2

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 199 Theorem 7.3.2. Let h : (0, 1) → R. Under the assumptions of Lemma 7.3.1, let ξ : [ϕ(c), ϕ(c) + η(ϕ(d), ϕ(c)) × Ω → R be Gh − HPηϕ SP with respect to η and ϕ, then the following inequality holds almost everywhere ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) η(ϕ(d), ϕ(c))   zϕ(c)(ϕ(c)+η(ϕ(d),ϕ(c)))  ϕ(c)+η(ϕ(d),ϕ(c)) ξ(z, .)ξ , . (z−ϕ(c))(ϕ(c)+η(ϕ(d),ϕ(c)))+zϕ(c) × dz z2 ϕ(c)  1 ≤ 2ξ(ϕ(c), .)ξ(ϕ(d), .) h2 (δ)dδ 0

2

2

+ [ξ (ϕ(c), .) + ξ (ϕ(d), .)]



1

0

h(δ)h(1 − δ)dδ.

Proof Since ξ is Gh − HPηϕ SP , therefore   ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ξ , . ≤ h(1 − δ)ξ(ϕ(c), .) + h(δ)ξ(ϕ(d), .), ϕ(c) + (1 − δ)η(ϕ(d), ϕ(c)) (7.8) and   ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ξ , . ≤ h(δ)ξ(ϕ(c), .) + h(1 − δ)ξ(ϕ(d), .), ϕ(c) + δη(ϕ(d), ϕ(c))

∀δ ∈ [0, 1]. (7.9)

From (7.8) and (7.9), we obtain     ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ,. ξ ,. ξ ϕ(c) + (1 − δ)η(ϕ(d), ϕ(c)) ϕ(c) + δη(ϕ(d), ϕ(c)) ≤ h2 (1 − δ)ξ(ϕ(c), .)ξ(ϕ(d), .) + h2 (δ)ξ(ϕ(c), .)ξ(ϕ(d), .) + h(δ)h(1 − δ)[ξ 2 (ϕ(c), .) + ξ 2 (ϕ(d), .)].

Integrating the above inequality with respect to δ over [0, 1], we have 

1 0

   ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ,. ξ , . dδ ξ ϕ(c) + (1 − δ)η(ϕ(d), ϕ(c)) ϕ(c) + δη(ϕ(d), ϕ(c))  1 ≤ ξ(ϕ(c), .)ξ(ϕ(d), .) [h2 (1 − δ) + h2 (δ)]dδ 

0

+ [ξ 2 (ϕ(c), .) + ξ 2 (ϕ(d), .)]



1

0

h(δ)h(1 − δ)dδ.

200

Integral Inequalities and Generalized Convexity  Set, z = ϕ(c)(ϕ(c)+η(ϕ(d),ϕ(c))) ϕ(c)+(1−δ)η(ϕ(d),ϕ(c)) , . in the left hand side of the above inequality, we obtain 

ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) η(ϕ(d), ϕ(c))   zϕ(c)(ϕ(c)+η(ϕ(d),ϕ(c)))  ϕ(c)+η(ϕ(d),ϕ(c)) ξ(z, .)ξ (z−ϕ(c))(ϕ(c)+η(ϕ(d),ϕ(c)))+zϕ(c) , . × dz z2 ϕ(c)  1 ≤ 2ξ(ϕ(c), .)ξ(ϕ(d), .) h2 (δ)dδ 0

+ [ξ 2 (ϕ(c), .) + ξ 2 (ϕ(d), .)]



1

0

h(δ)h(1 − δ)dδ.

This completes the proof. Theorem 7.3.3. Let h : (0, 1) → R. Under the assumptions of Lemma 7.3.1, let ξ, ψ : [ϕ(c), ϕ(c) + η(ϕ(d), ϕ(c)) × Ω → R be Gh − HPηϕ SP with respect to η and ϕ, then the following inequality holds almost everywhere  ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ϕ(c)+η(ϕ(d),ϕ(c)) ξ(z, .)ψ(z, .) dz η(ϕ(d), ϕ(c)) z2 ϕ(c)  1  1 2 ≤ M (c, d) h (δ)dδ + N (c, d) h(δ)h(1 − δ)dδ, 0

0

where, M (c, d) = ξ(ϕ(c), .)ψ(ϕ(c), .) + ξ(ϕ(d), .)ψ(ϕ(d), .) and N (c, d) = ξ(ϕ(c), .)ψ(ϕ(d), .) + ξ(ϕ(d), .)ψ(ϕ(c), .). Proof Since ξ and ψ are Gh − HPηϕ SP, we have   ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ξ , . ≤ h(1 − δ)ξ(ϕ(c), .) + h(δ)ξ(ϕ(d), .), ϕ(c) + (1 − δ)η(ϕ(d), ϕ(c)) (7.10) and ψ



 ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) , . ≤ h(1 − δ)ψ(ϕ(c), .) + h(δ)ψ(ϕ(d), .), ϕ(c) + (1 − δ)η(ϕ(d), ϕ(c))

∀ δ ∈ [0, 1].

(7.11)

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 201 Multiplying (7.10) and (7.11), it follows that     ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ξ ,. ψ ,. ϕ(c) + (1 − δ)η(ϕ(d), ϕ(c)) ϕ(c) + (1 − δ)η(ϕ(d), ϕ(c)) ≤ h2 (1 − δ)ξ(ϕ(c), .)ψ(ϕ(c), .) + h2 (δ)ξ(ϕ(d), .)ψ(ϕ(d), .)

+ h(δ)h(1 − δ)[ξ(ϕ(c), .)ψ(ϕ(d), .) + ξ(ϕ(d), .)ψ(ϕ(c), .)].

Integrating the above inequality over [0, 1] with respect to δ, we get 

1

   ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ,. ψ , . dδ ϕ(c) + (1 − δ)η(ϕ(d), ϕ(c)) ϕ(c) + (1 − δ)η(ϕ(d), ϕ(c))  1  1 ≤ ξ(ϕ(c), .)ψ(ϕ(c), .) h2 (1 − δ)dδ + ξ(ϕ(d), .)ψ(ϕ(d), .) h2 (δ)dδ ξ

0



0

+ [ξ(ϕ(c), .)ψ(ϕ(d), .) + ξ(ϕ(d), .)ψ(ϕ(c), .)]



0

1

h(δ)h(1 − δ)dδ.

0

This implies, 

ϕ(c)+η(ϕ(d),ϕ(c))

ξ(z, .)ψ(z, .) dz z2 ϕ(c)  1 ≤ [ξ(ϕ(c), .)ψ(ϕ(c), .) + ξ(ϕ(d), .)ψ(ϕ(d), .)] h2 (δ)dδ

ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) η(ϕ(d), ϕ(c))

0

+ [ξ(ϕ(c), .)ψ(ϕ(d), .) + ξ(ϕ(d), .)ψ(ϕ(c), .)] = M (c, d)



1

h2 (δ)dδ + N (c, d) 0



0

1



1

0

h(δ)h(1 − δ)dδ

h(δ)h(1 − δ)dδ.

This completes the proof. Lemma 7.3.2. Let ξ : K ⊂ R\{0} × Ω → R be a mean-square differentiable general stochastic process on K 0 and ϕ(c), ϕ(d) ∈ K 0 with ϕ(c) < ϕ(c) + η(ϕ(d), ϕ(c)). Then, we have almost everywhere ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c)))η(ϕ(d), ϕ(c)) 2    1 ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) (1 − 2δ)  × , . dδ ξ 2 ϕ(c) + δη(ϕ(d), ϕ(c)) 0 (ϕ(c) + δη(ϕ(d), ϕ(c))) 1 [ξ(ϕ(c), .) + ξ(ϕ(c) + η(ϕ(d), ϕ(c)), .)] 2  ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ϕ(c)+η(ϕ(d),ϕ(c)) ξ(z, .) dz. − η(ϕ(d), ϕ(c)) z2 ϕ(c)

=

(7.12)

202

Integral Inequalities and Generalized Convexity

Proof ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c)))η(ϕ(d), ϕ(c)) L.H.S. = 2    1 (1 − 2δ) ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c)))  , . dδ × ξ 2 ϕ(c) + δη(ϕ(d), ϕ(c)) 0 (ϕ(c) + δη(ϕ(d), ϕ(c)))  ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c)))η(ϕ(d), ϕ(c)) (1 − 2δ) = − − 2 (ϕ(c) + δη(ϕ(d), ϕ(c)))2 0   ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) , . dδ × ξ ϕ(c) + δη(ϕ(d), ϕ(c))   1  ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) 2δ − 1 ξ , .  = 2 ϕ(c) + δη(ϕ(d), ϕ(c)) 0   1  ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) , . dδ. − ξ ϕ(c) + δη(ϕ(d), ϕ(c)) 0 

Set z =

1

ϕ(c)(ϕ(c)+η(ϕ(d),ϕ(c))) ϕ(c)+δη(ϕ(d),ϕ(c)) ,

we obtain

1 [ξ(ϕ(c), .) + ξ(ϕ(c) + η(ϕ(d), ϕ(c)), .)] 2  ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ϕ(c)+η(ϕ(d),ϕ(c)) ξ(z, .) dz. − η(ϕ(d), ϕ(c)) z2 ϕ(c)

L.H.S. =

This completes the proof. Now we prove some integral inequalities with the help of Lemma 7.3.2. Theorem 7.3.4. Let h : (0, 1) → R and ξ : K × Ω ⊂ R\{0} × Ω → R be a mean-square differentiable general stochastic process on K 0 with respect to η and ϕ with ϕ(c) < ϕ(c) + η(ϕ(d), ϕ(c)). If |ξ  |q is general h−harmonically preinvex on [ϕ(c), ϕ(c) + η(ϕ(d), ϕ(c))] for q ≥ 1, then   ξ(ϕ(c), .) + ξ(ϕ(c) + η(ϕ(d), ϕ(c)), .)   2   ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ϕ(c)+η(ϕ(d),ϕ(c)) ξ(z, .)  − dz   η(ϕ(d), ϕ(c)) z2 ϕ(c) ≤

ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c)))η(ϕ(d), ϕ(c)) 2 q−1

1

× δ1 q [δ2 |ξ  (ϕ(c), .)|q + δ3 |ξ  (ϕ(d), .)|q ] q ,

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 203 where δ1 =



1

0

=

|1 − 2δ| dδ (ϕ(c) + δη(ϕ(d), ϕ(c)))2

1 ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) 2 log − 2 η (ϕ(d), ϕ(c))

δ2 =

δ3 =

 

1

|1 − 2δ|h(δ) dδ and (ϕ(c) + δη(ϕ(d), ϕ(c)))2

1

|1 − 2δ|h(1 − δ) dδ. (ϕ(c) + δη(ϕ(d), ϕ(c)))2

0

0

 2  ϕ(c) + 12 η(ϕ(d), ϕ(c)) , ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))

Proof Recall Lemma 7.3.2:   ξ(ϕ(c), .) + ξ(ϕ(c) + η(ϕ(d), ϕ(c)), .)   2 ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) − η(ϕ(d), ϕ(c))



ϕ(c)+η(ϕ(d),ϕ(c))

ϕ(c)

  ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c)))η(ϕ(d), ϕ(c)) =  2

×



≤ ×



1

0

(1 − 2δ) ξ (ϕ(c) + δη(ϕ(d), ϕ(c)))2



 ξ(z, .)  dz   z2

   ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) , . dδ  ϕ(c) + δη(ϕ(d), ϕ(c))

ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c)))η(ϕ(d), ϕ(c)) 2 1 0

         ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) (1 − 2δ)   ξ , .  dδ.  (ϕ(c) + δη(ϕ(d), ϕ(c)))2   ϕ(c) + δη(ϕ(d), ϕ(c))

204

Integral Inequalities and Generalized Convexity

Applying H¨ older’s inequality and the definition of Gh − HPηϕ SP in above inequality, we get   ξ(ϕ(c), .) + ξ(ϕ(c) + η(ϕ(d), ϕ(c)), .)   2   ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ϕ(c)+η(ϕ(d),ϕ(c)) ξ(z, .)  − dz   η(ϕ(d), ϕ(c)) z2 ϕ(c) ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c)))η(ϕ(d), ϕ(c)) 2  1  q−1 q |1 − 2δ| × dδ 2 0 (ϕ(c) + δη(ϕ(d), ϕ(c)))   q  q1  1    ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) |1 − 2δ|  , .  dδ ξ ×  2 ϕ(c) + δη(ϕ(d), ϕ(c)) 0 (ϕ(c) + δη(ϕ(d), ϕ(c))) ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c)))η(ϕ(d), ϕ(c)) ≤ 2  1  q−1 q |1 − 2δ| × dδ 2 0 (ϕ(c) + δη(ϕ(d), ϕ(c)))  1  q1 |1 − 2δ|  q  q × [h(δ)|ξ (ϕ(c), .)| + h(1 − δ)|ξ (ϕ(d), .)| ] 2 0 (ϕ(c) + δη(ϕ(d), ϕ(c))) ≤

=

ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c)))η(ϕ(d), ϕ(c)) 2 q−1

1

× δ1 q [δ2 |ξ  (ϕ(c), .)|q + δ3 |ξ  (ϕ(d), .)|q ] q . This completes the proof. Theorem 7.3.5. Let h : (0, 1) → R and ξ : K × Ω ⊂ R\{0} × Ω → R be a mean-square differentiable general stochastic process on K 0 with respect to η and ϕ with ϕ(c) < ϕ(c) + η(ϕ(d), ϕ(c)). If |ξ  |q is general h−harmonically preinvex on [ϕ(c), ϕ(c) + η(ϕ(d), ϕ(c))] for q > 1, p1 + 1q = 1 then   ξ(ϕ(c), .) + ξ(ϕ(c) + η(ϕ(d), ϕ(c)), .)   2   ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ϕ(c)+η(ϕ(d),ϕ(c)) ξ(z, .)  − dz   η(ϕ(d), ϕ(c)) z2 ϕ(c) ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c)))η(ϕ(d), ϕ(c)) 2  p1  1 1 [δ4 |ξ  (ϕ(c), .)|q + δ5 |ξ  (ϕ(d), .)|q ] q , × p+1

≤

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 205 where 1 δ4 = 0 1 δ5 = 0

h(δ) (ϕ(c)+δη(ϕ(d),ϕ(c)))2q dδ and h(1−δ) (ϕ(c)+δη(ϕ(d),ϕ(c)))2q dδ.

Proof From Lemma 7.3.2, we have   ξ(ϕ(c), .) + ξ(ϕ(c) + η(ϕ(d), ϕ(c)), .)   2   ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ϕ(c)+η(ϕ(d),ϕ(c)) ξ(z, .)  − dz   η(ϕ(d), ϕ(c)) z2 ϕ(c) ≤  ×

ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c)))η(ϕ(d), ϕ(c)) 2    1      ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) (1 − 2δ)  dδ.   ξ , .   (ϕ(c) + δη(ϕ(d), ϕ(c)))2   ϕ(c) + δη(ϕ(d), ϕ(c)) 0

Applying H¨ older’s inequality and the definition of Gh − HPηϕ SP in above inequality, we get   ξ(ϕ(c), .) + ξ(ϕ(c) + η(ϕ(d), ϕ(c)), .)   2   ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) ϕ(c)+η(ϕ(d),ϕ(c)) ξ(z, .)  − dz   η(ϕ(d), ϕ(c)) z2 ϕ(c)  1  p1 ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c)))η(ϕ(d), ϕ(c)) p |1 − 2δ| dδ ≤ 2 0   q  q1  1    ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c))) 1  , .  dδ ξ ×  2q ϕ(c) + δη(ϕ(d), ϕ(c)) 0 (ϕ(c) + δη(ϕ(d), ϕ(c)))   p1 1 ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c)))η(ϕ(d), ϕ(c)) ≤ 2 p+1  1  q1 1  q  q × [h(δ)|ξ (ϕ(c), .)| + h(1 − δ)|ξ (ϕ(d), .)| ] 2q 0 (ϕ(c) + δη(ϕ(d), ϕ(c))) ϕ(c)(ϕ(c) + η(ϕ(d), ϕ(c)))η(ϕ(d), ϕ(c)) = 2  p1  1 1 [δ4 |ξ  (ϕ(c), .)|q + δ5 |ξ  (ϕ(d), .)|q ] q . × p+1 This completes the proof.

206

Integral Inequalities and Generalized Convexity

7.4

Multidimensional General h−Harmonic Preinvex Stochastic Processes (M Gh − HPη ϕSP )

In this section, we give the definition of multidimensional general hharmonic preinvexity for stochastic processes with respect to η and ϕ on n coordinates and present Hermite–Hadamard type inequalities for (M Gh − HPη ϕSP ) [155]. Definition 7.4.1. Let h : (0, 1) → R be a nonnegative function. A stochastic process ξ : Kηϕ (⊂ R+ n ) × Ω → R is said to be M Gh − HPηϕ SP with respect to η and ϕ on Kηϕ , if the following inequality holds almost everywhere   ϕ(x1 )(ϕ(x1 ) + η(ϕ(y1 ), ϕ(x1 ))) ϕ(xn )(ϕ(xn ) + η(ϕ(yn ), ϕ(xn ))) ξ , ..., ,. ϕ(x1 ) + δη(ϕ(y1 ), ϕ(x1 )) ϕ(xn ) + δη(ϕ(yn ), ϕ(xn )) ≤ h(δ)ξ(ϕ(x1 ), ..., ϕ(xn ), .) + h(1 − δ)ξ(ϕ(y1 ), ..., ϕ(yn ), .),

for all (ϕ(x1 ), ..., ϕ(xn )), (ϕ(y1 ), ..., ϕ(yn )) ∈ Kηϕ and δ ∈ [0, 1]. We now discuss some special cases of Definition 7.4.1: 1 If h(δ) = δ, then above definition reduces to the definition of multidimensional general harmonic preinvex stochastic process with respect to η and ϕ on Kηϕ . 2 If h(δ) = δ s , then above definition reduces to the definition of Breckner type of multidimensional general s−harmonic preinvex stochastic process with respect to η and ϕ on Kηϕ . 3 If h(δ) = δ −s , then above definition reduces to the definition of Godunova-Levin type of multidimensional general s−harmonic preinvex stochastic process with respect to η and ϕ on Kηϕ . 4 If h(δ) = 1, then above definition reduces to the definition of multidimensional general harmonic P − preinvex stochastic process with respect to η and ϕ on Kηϕ . Definition 7.4.2. Let h : (0, 1) → (0, ∞). A stochastic process ξ : Kηϕ (⊂ R+ n ) × Ω → R is called M Gh − HPηϕ SP with respect to η and ϕ on i n−coordinates if the following stochastic mappings ξϕ(x : K × Ω → R are n) general h−harmonically preinvex with respect to η and ϕ on K almost everywhere for all ϕ(x) ∈ K :    i n ξϕ(x (ϕ(x), .) = ξ ∧i−1 k=1 ϕ(xk ), ϕ(x), ∧k=i+1 ϕ(xk ) , . . n)

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 207 Lemma 7.4.1. Every M Gh − HPηϕ SP is Gh − HPηϕ SP with respect to η and ϕ on n− coordinates almost everywhere, but not conversely. Proof Suppose ξ : Kηϕ × Ω → R be M Gh − HPηϕ SP with respect to η and ϕ on Kηϕ . i : K × Ω → R by Define ξϕ(x n) i n ξϕ(xn ) (ϕ(x), .) = ξ((∧i−1 k=1 ϕ(xk ), ϕ(x), ∧k=i+1 ϕ(xk )), .) almost everywhere for ϕ(x) ∈ K. Then for ϕ(x), ϕ(y) ∈ K and δ ∈ [0, 1] almost everywhere   ϕ(x)(ϕ(x) + η(ϕ(y), ϕ(x))) i ξϕ(x , . n) ϕ(x) + (1 − δ)η(ϕ(y), ϕ(x))    ϕ(x)(ϕ(x) + η(ϕ(y), ϕ(x))) n =ξ ∧i−1 , ∧ ϕ(x ), ϕ(x ) ,. k k k=1 ϕ(x) + (1 − δ)η(ϕ(y), ϕ(x)) k=i+1 n ≤ h(1 − δ)ξ((∧i−1 k=1 ϕ(xk ), ϕ(x), ∧k=i+1 ϕ(xk )), .) n + h(δ)ξ((∧i−1 k=1 ϕ(xk ), ϕ(y), ∧k=i+1 ϕ(xk )), .)

i i = h(1 − δ)ξϕ(x (ϕ(x), .) + h(δ)ξϕ(x (ϕ(y), .). n) n)

For converse part, we give the following example. Example 7.4.1. Let ξ : Kηϕ = [1, 2] × [2, 3] × [3, 4] × Ω → R be a stochastic process defined as ξ(ϕ(x1 ), ϕ(x2 ), ϕ(x3 ), .) = (ϕ(x1 ) − 1)(ϕ(x2 ) − 2)(ϕ(x3 ) − 3) and η : Kηϕ × Kηϕ → R, η(ϕ(x), ϕ(y)) = ϕ(x) − ϕ(y), ∀ϕ(x), ϕ(y) ∈ Kηϕ . If we take ϕ(x) = (1, 3, 4), ϕ(y) = (2, 2, 4) and h(δ) = δ. Then, 

ϕ(x1 )(ϕ(x1 ) + η(ϕ(y1 ), ϕ(x1 ))) ϕ(x2 )(ϕ(x2 ) + η(ϕ(y2 ), ϕ(x2 ))) , , ϕ(x1 ) + δη(ϕ(y1 ), ϕ(x1 )) ϕ(x2 ) + δη(ϕ(y2 ), ϕ(x2 ))  ϕ(x3 )(ϕ(x3 ) + η(ϕ(y3 ), ϕ(x3 ))) ,. ϕ(x3 ) + δη(ϕ(y3 ), ϕ(x3 ))  ϕ(x1 )ϕ(y1 ) ϕ(x2 )ϕ(y2 ) =ξ , , (1 − δ)ϕ(x1 ) + δϕ(y1 ) (1 − δ)ϕ(x2 ) + δϕ(y2 )  ϕ(x3 )ϕ(y3 ) ,. (1 − δ)ϕ(x3 ) + δϕ(y3 )    6 2δ(1 − δ) 2 −1 −2 = > 0, (if δ = 0) (7.13) = 1+δ 3−δ (1 + δ)(3 − δ)

ξ

and h(δ)ξ(ϕ(x1 ), ϕ(x2 ), ϕ(x3 ), .) + h(1 − δ)ξ(ϕ(y1 ), ϕ(y2 ), ϕ(y3 ), .) = δ.0 + (1 − δ).0 = 0.

(7.14)

208

Integral Inequalities and Generalized Convexity

From (7.13) and (7.14), we get  ϕ(x1 )(ϕ(x1 ) + η(ϕ(y1 ), ϕ(x1 ))) ϕ(x2 )(ϕ(x2 ) + η(ϕ(y2 ), ϕ(x2 ))) , , ξ ϕ(x1 ) + δη(ϕ(y1 ), ϕ(x1 )) ϕ(x2 ) + δη(ϕ(y2 ), ϕ(x2 ))  ϕ(x3 )(ϕ(x3 ) + η(ϕ(y3 ), ϕ(x3 ))) ,. ϕ(x3 ) + δη(ϕ(y3 ), ϕ(x3 )) > h(δ)ξ(ϕ(x1 ), ϕ(x2 ), ϕ(x3 ), .) + h(1 − δ)ξ(ϕ(y1 ), ϕ(y2 ), ϕ(y3 ), .),

which is contradiction. This completes the proof. Consider n−dimensional interval ♦n = Πni=1 n = Πni=1 [ϕ(wi ), ϕ(wi ) + η(ϕ(zi ), ϕ(wi ))] ⊆ Rn+ . For simplicity, let i1 = ϕ(wi ) and i2 = ϕ(wi ) + η(ϕ(zi ), ϕ(wi )) with η(ϕ(zi ), ϕ(wi ) > 0 for each i = 1, 2, ..., n. Remark 7.4.1. Let h : (0, 1) → R and ξ : ♦n × Ω → R+ is M Gh − HPηϕ SP with respect to η and ϕ on ♦n . If the assumptions of Lemma 7.3.1 satisfy, i then ξϕ(x : [ϕ(wi ), ϕ(wi ) + η(ϕ(zi ), ϕ(wi ))] × Ω → R is Gh − HPηϕ SP on n) [ϕ(wi ), ϕ(wi ) + η(ϕ(zi ), ϕ(wi ))] with respect to η and ϕ for each i = 1, 2, ..., n. Thus    i2 ξ i i1 i2 2 i1 i2 1 ϕ(xn ) (ϕ(xi ), .) i ξ dϕ(xi ) , . ≤ ϕ(xn ) 1 i i i i ϕ2 (xi ) 1 + 2 2 − 1 i1 2h( 2 )  1 i i ≤ [ξϕ(xn ) (ϕ(wi ), .) + ξϕ(xn ) (ϕ(zi ), .)] h(δ)dδ, 0

almost everywhere for i = 1, 2, ..., n.

Theorem 7.4.1. Let h : (0, 1) → R and ξ : ♦n ×Ω → R+ be M Gh−HPηϕ SP with respect to η and ϕ on ♦n . If the assumptions of Lemma 7.3.1 satisfy, then almost everywhere   n−1 i+1   2 i1 i2 2 i+1 1 i−1 n 1 2 ξ ∧ ϕ(x ), , , ∧ ϕ(x ) ,. k k k=i+2 k=1 i1 + i2 i+1 4h2 ( 12 ) i=1 + i+1 1 2  i+1 i+1  21 2  i2 ξ i+1 n−1 ,. ϕ(xn ) i+1 1  i1 i2 +i+1 1 2 dϕ(xi ) ≤ ϕ2 (xi ) 2h( 12 ) i=1 i2 − i1 i1 i+1 n−1 i+1  i2  i+1  i i ξϕ(x (ϕ(xi+1 ), .) 2 i+1 n) 1 2 1 2 dϕ(xi+1 )dϕ(xi ) ≤ ϕ2 (xi+1 )ϕ2 (xi ) i2 − i1 i+1 − i+1 i1 i+1 2 1 1 i=1   1 h(δ)dδ ≤ 0   i+1 i+1 n−1  i i  i2 ξϕ(x (ϕ(w ), .) + ξ (ϕ(z ), .) i+1 i+1 ) ) ϕ(x n n 1 2 × dϕ(xi ) i − i 2 (x ) i ϕ  i  2 1 1 i=1

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 209 ≤



+ξ +ξ +ξ

1

h(δ)dδ 0

2 n−1      n ξ ∧i−1 k=1 ϕ(xk ), ϕ(wi ), ϕ(wi+1 ), ∧k=i+2 ϕ(xk ) , .

i=1   i−1 ∧k=1 ϕ(xk ), ϕ(zi ), ϕ(wi+1 ), ∧nk=i+2 ϕ(xk ) , .    i−1 ∧k=1 ϕ(xk ), ϕ(wi ), ϕ(zi+1 ), ∧nk=i+2 ϕ(xk ) , .





  n ∧i−1 k=1 ϕ(xk ), ϕ(zi ), ϕ(zi+1 ), ∧k=i+2 ϕ(xk ) , . .

(7.15)

Proof Recall Remark 7.4.1:   i+1 1 2 i+1 i+1 1 2 ξ , . 2h( 12 ) ϕ(xn ) i+1 + i+1 1 2 i+1 i+1 i+1  i+1 ξ 2  1 2 ϕ(xn ) (ϕ(xi+1 ), .) ≤ i+1 dϕ(xi+1 ) ϕ2 (xi+1 ) 2 − i+1 i+1 1 1  1 i+1 i+1 ≤ [ξϕ(x (ϕ(w ), .) + ξ (ϕ(z ), .)] h(δ)dδ, almost everywhere. i+1 i+1 ϕ(xn ) n) 0

Integrating above inequality on i with respect to ϕ(xi ), we have  i+1 i+1  21 2  i2 ξ i+1 ,. i i ϕ(xn ) i+1 1 1 2 +i+1 1 2 dϕ(xi ) 1 i i 2 ϕ (xi ) 2h( 2 ) 2 − 1 i1 i+1 i+1  i2  i+1 ξϕ(x (ϕ(xi+1 ), .) 2 i1 i2 i+1 n) 1 2 dϕ(xi+1 )dϕ(xi ) ≤ i ϕ2 (xi+1 )ϕ2 (xi ) 2 − i1 i+1 − i+1 i1 i+1 2 1 1 ≤



1

h(δ)dδ 0

i  i × i 1 2i 2 − 1





i2

i1



i+1 i+1 ξϕ(x (ϕ(wi+1 ), .) + ξϕ(x (ϕ(zi+1 ), .) n) n)

ϕ2 (xi )



dϕ(xi ),

almost everywhere. Applying Hermite–Hadamard integral inequality, we have    i+1 2 i1 i2 2 i+1 1 i−1 n 1 2 ξ ∧k=1 ϕ(xk ), i , , ∧k=i+2 ϕ(xk ) , . 1 + i2 i+1 4h2 ( 12 ) + i+1 1 2  i+1 i+1  21 2  i2 ξ i+1 ,. ϕ(xn ) i+1 1 i1 i2 +i+1 1 2 ≤ dϕ(xi ) ϕ2 (xi ) 2h( 12 ) i2 − i1 i1 i+1 i+1  i2  i+1 ξϕ(x (ϕ(xi+1 ), .) 2  i i i+1 n) 1 2 dϕ(xi+1 )dϕ(xi ) ≤ i 1 2 i i+1 i+1 2 ϕ (xi+1 )ϕ2 (xi ) 2 − 1 2 − 1 i1 i+1 1   1  i2  ξ i+1 (ϕ(w ), .) + ξ i+1 (ϕ(z ), .)  i+1 i+1 i1 i2 ϕ(xn ) ϕ(xn ) dϕ(xi ) h(δ)dδ ≤ i − i 2 (x ) i ϕ  i 0 2 1 1

210

Integral Inequalities and Generalized Convexity 

2 h(δ)dδ 0   i−1   × ξ ∧k=1 ϕ(xk ), ϕ(wi ), ϕ(wi+1 ), ∧nk=i+2 ϕ(xk ) , .    n + ξ ∧i−1 k=1 ϕ(xk ), ϕ(zi ), ϕ(wi+1 ), ∧k=i+2 ϕ(xk ) , .    n + ξ ∧i−1 k=1 ϕ(xk ), ϕ(wi ), ϕ(zi+1 ), ∧k=i+2 ϕ(xk ) , .    n + ξ ∧i−1 k=1 ϕ(xk ), ϕ(zi ), ϕ(zi+1 ), ∧k=i+2 ϕ(xk ) , . , almost everywhere,

≤

1

where i = 1, 2, ..., n − 1. Taking summation from 1 to n − 1, we compute the result easily.

Remark 7.4.2. If h(δ) = δ and ξ is M GPηϕ SP , then above theorem reduces to Theorem 2.2 of [129], i.e.

ξ



  + i+1 i1 + i2 i+1 2 , 1 , ∧nk=i+2 ϕ(xk ) , . 2 2 i=1 i    n−1 2  i+1 1 + i+1 i+1 1 2 , . dϕ(xi ) ≤ ξ 2 i2 − i1 i1 ϕ(xn ) i=1  i2  i+1 n−1  2 1 i+1 ≤ ξϕ(x (ϕ(xi+1 ), .)dϕ(xi+1 )dϕ(xi ) i+1 i+1 i i n) i+1 i ( −  )( −  ) 1 1 2 1 2 1 i=1 ≤

n−1 

n−1 

∧i−1 k=1 ϕ(xk ),

i=1

1 2(i2 − i1 )



i2 i1



 i+1 i+1 ξϕ(x (ϕ(w ), .) + ξ (ϕ(z ), .) dϕ(xi ) i+1 i+1 ϕ(xn ) n)

n−1   1    i−1 ξ ∧k=1 ϕ(xk ), ϕ(wi ), ϕ(wi+1 ), ∧nk=i+2 ϕ(xk ) , . 4 i=1    n + ξ ∧i−1 k=1 ϕ(xk ), ϕ(zi ), ϕ(wi+1 ), ∧k=i+2 ϕ(xk ) , .    n + ξ ∧i−1 k=1 ϕ(xk ), ϕ(wi ), ϕ(zi+1 ), ∧k=i+2 ϕ(xk ) , .    n + ξ ∧i−1 k=1 ϕ(xk ), ϕ(zi ), ϕ(zi+1 ), ∧k=i+2 ϕ(xk ) , . .

≤

Theorem 7.4.2. Let h : (0, 1) → R and ξ : ♦n ×Ω → R+ be M Gh−HPηϕ SP with respect to η and ϕ on ♦n . If the assumptions of Lemma 7.3.1 satisfy, then almost everywhere   i2  ξ i n i  i1 i2 ϕ(wn ) (ϕ(xi ), .) + ξϕ(zn ) (ϕ(xi ), .) dϕ(xi ) ϕ2 (xi ) i2 − i1 i1 i=1 ≤ [n(ξ(ϕ(w), .) + ξ(ϕ(z), .)) + i (ϕ(wi ), .))] + ξϕ(z n)



1

n  i=1

h(δ)dδ. 0

i (ξϕ(w (ϕ(zi ), .) n)

(7.16)

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 211 i i Proof Applying Remark 7.4.1 for ξϕ(w and ξϕ(z , i = 1, 2, ..., n, respecn) n) tively, then almost everywhere

i1 i2 i2 − i1



i2

i ξϕ(w (ϕ(xi ), .) n)

ϕ2 (xi )

i1

dϕ(xi )

i i ≤ [ξϕ(w (ϕ(wi ), .) + ξϕ(w (ϕ(zi ), .)] n) n)

= [ξ(ϕ(w), .) +

i ξϕ(w (ϕ(zi ), .)] n)



1



1

h(δ)dδ 0

h(δ)dδ,

(7.17)

0

and i1 i2 i2 − i1



i2

i ξϕ(z (ϕ(xi ), .) n)

ϕ2 (xi )

i1

dϕ(xi )

i i ≤ [ξϕ(z (ϕ(wi ), .) + ξϕ(z (ϕ(zi ), .)] n) n) i = [ξϕ(z (ϕ(wi ), .) + ξ(ϕ(z), .)] n)



1



1

h(δ)dδ 0

h(δ)dδ.

Adding (7.17) and (7.18), we obtain   i2  ξ i i i1 i2 ϕ(wn ) (ϕ(xi ), .) + ξϕ(zn ) (ϕ(xi ), .) dϕ(xi ) ϕ2 (xi ) i2 − i1 i1 ≤ [ξ(ϕ(w), .) +

i ξϕ(w (ϕ(zi ), .) n)

+

(7.18)

0

i ξϕ(z (ϕ(wi ), .) n)

+ ξ(ϕ(z), .)]

almost everywhere,



1

h(δ)dδ, 0

where i = 1, 2, ..., n. Taking summation from 1 to n, we compute the result easily. Remark 7.4.3. If h(δ) = δ and ξ is M GPηϕ SP , then above theorem reduces to Theorem 2.3 of [129], i.e. n  i=1

≤

1 2(i2 − i1 )



i2 i1

 i i (ϕ(xi ), .) + ξϕ(z (ϕ(xi ), .) dϕ(xi ) ξϕ(w n) n)



n

n 1 i [ξ(ϕ(w), .) + ξ(ϕ(z), .)] + [ξ (ϕ(zi ), .) 2 2 i=1 ϕ(wn )

i (ϕ(wi ), .)]. + vϕ(z n)

Theorem 7.4.3. Let h : (0, 1) → R and ξ : ♦n ×Ω → R+ be M Gh−HPηϕ SP with respect to η and ϕ on ♦n . If the assumptions of Lemma 7.3.1 satisfy, then

212

Integral Inequalities and Generalized Convexity

almost everywhere    2 n1 n2 2 11 12 2 21 22 1 ξ ,. , , ..., n 11 + 12 21 + 22 1 + n2 2n hn ( 12 )  12  22  n2 ξ n n  i1 i2 ϕ(xn ) (ϕ(xn ), .) n ... dϕ(xn )dϕ(xn−1 )...dϕ(x1 ) ≤ i i 2 1 2 n  − 1 1 1 1 i=1 ϕ (xi ) i=1 2 n+1   1 h(δ)dδ [ξ((h(ρ)ϕ(w) + h(1 − ρ)ϕ(z)), .)], ≤ 0

ρ∈mi (n)

where mi (n) = {ρ = (ρ1 , ρ2 , ..., ρn ) ∈ Nn0 : ρi ≤ 1; |ρ| = n + 1 − i; i = 1, 2, ..., n + 1}; |ρ| : ρ1 + ρ2 + ... + ρn ; ρϕ(w) = (ρ1 ϕ(w1 ), ρ2 ϕ(w2 ), ..., ρn ϕ(wn )). Proof Recall Remark 7.4.1:    n2 ξ n n1 n2 2 n1 n2 1 ϕ(xn ) (ϕ(xn ), .) n dϕ(xn ) 1 ξϕ(xn ) n + n , . ≤ n − n n  ϕ2 (xn ) 2h( 2 ) 1 2 2 1 1  1 n n ≤ [ξϕ(x (ϕ(w ), .) + ξ (ϕ(z ), .)] h(δ)dδ, almost everywhere. n n ϕ(xn ) n) 0

Integrating above inequality on  n−1 1 n−1 1 2 1 n−1 2h( 2 ) 2 − n−1 1

n−1

, we obtain  n n  21 2 n  n−1 ξϕ(x 2 n +n , . n) 1

n−1 1

2

ϕ2 (xn−1 )

dϕ(xn−1 )

n−1 n−1 n1 n2 1 2 n−1 n−1 n − n 2 − 1 2 1  n2 ξ n  n−1 2 (ϕ(x n ), .) ϕ(xn ) dϕ(xn )dϕ(xn−1 ) × 2 2 ϕ (xn )ϕ (xn−1 ) n−1 n 1 1  1  n−1 n−1 1 2 ≤ h(δ)dδ n−1 n−1 −  0 2 1 n−1  2 ξ n n (ϕ(w ), .) + ξϕ(x (ϕ(zn ), .) n ϕ(xn ) n) dϕ(xn−1 ), × 2 n−1 ϕ (xn−1 ) 1 ≤

almost everywhere. Using Hermite–Hadamard integral inequality, we have    1 2 n−1 2 n1 n2 n−1 n−2 1 2 ξ ∧k=1 ϕ(xk ), n−1 , n ,. 1 + n2 4h2 ( 12 ) 1 + n−1 2  n n  21 2 n n−1  n−1 ξϕ(x n,. 2 ) n 1  n−1 n 1 +2 1 2 dϕ(xn−1 ) ≤ ϕ2 (xn−1 ) 2h( 12 ) n−1 − n−1 n−1 2 1 1

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 213 n−1 n1 n2 n−1 1 2 n−1 n−1 n − n 2 − 1 2 1  n2 ξ n  n−1 2 (ϕ(x n ), .) ϕ(xn ) dϕ(xn )dϕ(xn−1 ) × 2 2 ϕ (xn )ϕ (xn−1 ) n−1 n 1 1  1  n−1 n−1 1 2 ≤ n−1 h(δ)dδ 2 − n−1 0 1  n−1 n n 2 ξϕ(xn ) (ϕ(wn ), .) + ξϕ(x (ϕ(zn ), .) n) × dϕ(xn−1 ) 2 ϕ (xn−1 ) n−1 1  1 2   n−2   ≤ ξ ∧k=1 ϕ(xk ), ϕ(wn−1 ), ϕ(wn ) , . h(δ)dδ 0    + ξ ∧n−2 k=1 ϕ(xk ), ϕ(zn−1 ), ϕ(wn ) , .    +ξ ∧n−2 k=1 ϕ(xk ), ϕ(wn−1 ), ϕ(zn ) , .    + ξ ∧n−2 k=1 ϕ(xk ), ϕ(zn−1 ), ϕ(zn ) , . , almost everywhere.

≤

Integrating above inequality on n−2 , we get

n−2 1 n−2 1 2 n−2 4h2 ( 12 ) n−2 −  2 1    n 2n−1 n−1 2n 1 2 1 2  n−2 ξ ∧n−2 ϕ(x ), , n n n−1 n−1 k 2 k=1 1 +2 , . 1 +2 dϕ(xn−2 ) × 2 ϕ (xn−2 ) n−2 1 n−2 n−2 n−1 n−1 n1 n2 1 2 1 2 n−2 n−2 n−1 n−1 n − n 2 − 1 2 − 1 2 1  n−1  n2  n−2 2 2 ξn (ϕ(x ), .) ≤

× ≤ ×

n−2 1





n−1 1

1

h(δ)dδ

0

n−2 2

n−2 1

+ξ +ξ +ξ



2

n 1

n

ϕ(xn ) )ϕ2 (x

n

n−1 )ϕ

n−2 )

dϕ(xn )dϕ(xn−1 )dϕ(xn−2 )

n−2 )

  n−2   ξ ∧k=1 ϕ(xk ), ϕ(wn−1 ), ϕ(wn ) , .  

∧n−2 k=1 ϕ(xk ), ϕ(zn−1 ), ϕ(wn ) , .    n−2 ∧k=1 ϕ(xk ), ϕ(wn−1 ), ϕ(zn ) , . 

2 (x

n−2 n−2 1 2 n−2 2 − n−2 1

1 ϕ2 (x

ϕ2 (x

  ∧n−2 k=1 ϕ(xk ), ϕ(zn−1 ), ϕ(zn ) , . dϕ(xn−2 ), almost everywhere.

214

Integral Inequalities and Generalized Convexity

Using Hermite–Hadamard integral inequality in above inequality, we obtain    2 n−2 2 n−1 2 n1 n2 1 n−2 n−1 n−3 1 2 1 2 ξ ∧ ϕ(x ), , , ,. k k=1 n1 + n2 8h3 ( 12 ) n−2 + n−2 n−1 + n−1 1 2 1 2 1 n−2 n−2 1 2 4h2 ( 12 ) n−2 − n−2 2 1    n 2n−1 n−1 2n 1 2 1 2  n−2 ,. ξ ∧n−2 n 2 k=1 ϕ(xk ), n−1 +n−1 , n + 1 2 1 2 dϕ(xn−2 ) × ϕ2 (xn−2 ) n−2 1

≤

n−2 n−1 n−1 n1 n2 n−2 1 2 1 2 n2 − n1 n−2 − n−2 n−1 − n−1 2 1 2 1  n−1  n2  n−2 n 2 2 ξϕ(x (ϕ(xn ), .) n) dϕ(xn )dϕ(xn−1 )dϕ(xn−2 ) × 2 (x )ϕ2 (x 2 n−2 n−1 n ϕ n n−1 )ϕ (xn−2 ) 1 1 1

≤

≤



×



1

h(δ)dδ 0 n−2 2

n−2 1

2

n−2 n−2 1 2 n−2 n−2 −  2 1

1 ϕ2 (x

n−2 )

  n−2   ξ ∧k=1 ϕ(xk ), ϕ(wn−1 ), ϕ(wn ) , .



  ∧n−2 k=1 ϕ(xk ), ϕ(zn−1 ), ϕ(wn ) , .    + ξ ∧n−2 k=1 ϕ(xk ), ϕ(wn−1 ), ϕ(zn ) , .    + ξ ∧n−2 k=1 ϕ(xk ), ϕ(zn−1 ), ϕ(zn ) , . dϕ(xn−2 )

+ξ

≤



+ + + + +

h(δ)dδ

0

3

    n−3 ξ ∧k=1 ϕ(xk ), ϕ(wn−2 ), ϕ(wn−1 ), ϕ(wn ) , .

  ∧n−3 k=1 ϕ(xk ), ϕ(zn−2 ), ϕ(wn−1 ), ϕ(wn ) , .    ξ ∧n−3 k=1 ϕ(xk ), ϕ(wn−2 ), ϕ(zn−1 ), ϕ(wn ) , .    ξ ∧n−3 k=1 ϕ(xk ), ϕ(zn−2 ), ϕ(zn−1 ), ϕ(wn ) , .    ξ ∧n−3 k=1 ϕ(xk ), ϕ(wn−2 ), ϕ(wn−1 ), ϕ(zn ) , .    ξ ∧n−3 k=1 ϕ(xk ), ϕ(zn−2 ), ϕ(wn−1 ), ϕ(zn ) , .    ξ ∧n−3 k=1 ϕ(xk ), ϕ(wn−2 ), ϕ(zn−1 ), ϕ(zn ) , .    ξ ∧n−3 k=1 ϕ(xk ), ϕ(zn−2 ), ϕ(zn−1 ), ϕ(zn ) , . , almost everywhere.

+ξ +

1



Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 215 Doing this procedure successively, we obtain    1 2 n1 n2 2 11 12 2 21 22 ξ ,. , , ..., 11 + 12 21 + 22 n1 + n2 2n hn ( 21 ) ≤ ≤

 12  22  n2 ξ n n  i1 i2 ϕ(xn ) (ϕ(xn ), .) n ... dϕ(xn )dϕ(xn−1 )...dϕ(x1 ) i i 2  − 1 11 21 n i=1 ϕ (xi ) 1 i=1 2 

1

h(δ)dδ 0

n+1



ρ∈mi (n)

[ξ((h(ρ)ϕ(w) + h(1 − ρ)ϕ(z)), .)].

This completes the proof. Remark 7.4.4. If h(δ) = δ and ξ is M GPηϕ SP , then above theorem reduces to Theorem 2.4 of [129], i.e.    1 n1 + n2 1 + 12 21 + 22 , , ..., ,. ξ 2 2 2 1 i − i ) ( 2 1 i=1  12  22  ... ×

≤ n

11

≤

1 2n

n 1

21



ρ∈mi (n)

n 2

ξ((ϕ(x1 ), ..., ϕ(xn )).)dϕ(xn )dϕ(xn−1 )...dϕ(x1 )

[ξ((ρϕ(w) + (1 − ρ)ϕ(z)), .)].

Example 7.4.2. Let ξ : ♦2 ×Ω → R+ be two dimensional Gh−HPηϕ SP with respect to η and ϕ on ♦2 . Then from Theorem 7.4.3 for n = 2 and h(δ) = δ, we get    2 11 12 2 21 22 , ,. ξ 11 + 12 21 + 22  12  22 ξ 2 21 22 11 12 ϕ(x2 ) (ϕ(x2 ), .) dϕ(x2 )dϕ(x1 ) ≤ 1 1 2 2 2 − 1 2 − 1 11 21 ϕ2 (x1 )ϕ2 (x2 ) ≤

1 8



ρ∈mi (2)

[ξ((ρϕ(w) + (1 − ρ)ϕ(z)), .)],

where mi (2) = {ρ = (ρ1 , ρ2 ) ∈ N20 : ρi ≤ 1; |ρ| = n + 1 − i; i = 1, 2, 3}; |ρ| : ρ1 + ρ2 ; ρϕ(w) = (ρ1 ϕ(w1 ), ρ2 ϕ(w2 )). Therefore, m1 (2) = {(1, 1)}, m2 (2) = {(0, 1), (1, 0)} and m3 (2) = {(0, 0)}.

216

Integral Inequalities and Generalized Convexity Now using these values in above inequality, we compute    2 11 12 2 21 22 ,. ξ , 11 + 12 21 + 22  12  22 ξ 2 21 22 11 12 ϕ(x2 ) (ϕ(x2 ), .) dϕ(x2 )dϕ(x1 ) ≤ 1 2 − 11 22 − 21 11 21 ϕ2 (x1 )ϕ2 (x2 ) 1 ≤ [ξ((ϕ(w1 ), ϕ(w2 )), .) + ξ((ϕ(z1 ), ϕ(w2 )), .) + ξ((ϕ(w1 ), ϕ(z2 )), .) 8 + ξ((ϕ(z1 ), ϕ(z2 )), .)].

7.5

Strongly Generalized Convex Stochastic Processes

In this section, we introduce the concept of strongly η-convex stochastic processes [156]. Definition 7.5.1. Let (Ω, A, P ) be a probability space and K ⊆ R be an interval. Let µ(.) denote a positive random variable, then ξ : K ×Ω → R is said to be strongly η-convex stochastic process with respect to η : ξ(K) × ξ(K) → R and modulus µ(.) > 0 if ξ(δx + (1 − δ)y, .) ≤ ξ(y, .) + δη(ξ(x, .), ξ(y, .)) − µ(.)δ(1 − δ)(d − c)2 (a.e.) ∀x, y ∈ K and δ ∈ [0, 1]. Remark 7.5.1. If η(ξ(x, .), ξ(y, .)) = ξ(x, .) − ξ(y, .), then the definition of strongly η-convex stochastic process reduces to the definition of strongly convex stochastic process proposed by Kotrys [88]. When µ(.) = 0, then above definition reduces to the definition of η-convex stochastic process [69]. Example 7.5.1. Let ξ : (0, ∞) × Ω → R be a stochastic process defined as ξ(u, .) = u, and η : ξ((0, ∞)) × ξ((0, ∞)) → R, η(ξ(u, .), ξ(v, .)) = (ξ(u, .) − ξ(v, .))2 + ξ(u, .) + ξ(v, .). Then ξ is strongly η-convex stochastic processes with modulus 1. Theorem 7.5.1. A random variable ξ : K × Ω → R is an strongly η-convex stochastic process with modulus µ(.) > 0 if and only if for any κ1 , κ2 , κ3 ∈ K with κ1 ≤ κ2 ≤ κ3 , we have 0 (κ3 − κ2 ) (κ3 − κ1 )

1 ξ(κ2 , .) − ξ(κ3 , .) η(ξ(κ1 , .), ξ(κ3 , .))

1 0 ≥ 0. µ(.)(κ2 − κ1 )(κ3 − κ1 )

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 217 Proof Suppose that ξ is an strongly η-convex stochastic process and κ1 , κ2 , κ3 ∈ K with κ1 ≤ κ2 ≤ κ3 . Then there exist δ ∈ (0, 1), such that κ2 = δκ1 + (1 − δ)κ3 .   κ 2 − κ3 η(ξ(κ1 , .), ξ(κ3 , .)) ξ(κ2 , .) = ξ(δκ1 + (1 − δ)κ3 , .) ≤ ξ(κ3 , .) + κ1 − κ3    κ1 − κ 2 κ2 − κ3 (κ3 − κ1 )2 . − µ(.) κ1 − κ3 κ1 − κ3 This implies,

(ξ(κ3 , .) − ξ(κ2 , .))(κ3 − κ1 ) + (κ3 − κ2 )η(ξ(κ1 , .), ξ(κ3 , .)) − µ(.)(κ3 − κ2 )(κ2 − κ1 )(κ3 − κ1 ) ≥ 0,

0 or (κ3 − κ2 ) (κ3 − κ1 )

1 ξ(κ2 , .) − ξ(κ3 , .) η(ξ(κ1 , .), ξ(κ3 , .))

1 0 ≥ 0. µ(.)(κ2 − κ1 )(κ3 − κ1 )

For the converse part, take x1 , x2 ∈ I with x1 ≤ x2 . Choose any δ ∈ (0, 1), then we have x1 ≤ δx1 + (1 − δ)x2 ≤ x2 . The above determinant is 0 1 1 (x2 − (δx1 + (1 − δ)x2 )) ξ(δx1 + (1 − δ)x2 , .) − ξ(x2 , .) 0 (x2 − x1 ) η(ξ(x1 , .), ξ(x2 , .)) µ(.)(δx1 + (1 − δ)x2 − x1 )(x2 − x1 ) ≥ 0.

This implies ξ(δx1 + (1 − δ)x2 , .) ≤ ξ(x2 , .) + δη(ξ(x1 , .), ξ(x2 , .)) − µ(.)δ(1 − δ)(x2 − x1 )2 . Remark 7.5.2. When µ(.) = 0, then above theorem reduces to Theorem 1.10 of [69], i.e. (κ3 − κ2 ) (κ3 − κ1 )

ξ(κ2 , .) − ξ(κ3 , .) ≥ 0. η(ξ(κ1 , .), ξ(κ3 , .))

Now, we present Hermite–Hadamard inequality, Ostrowski inequality and some other interesting inequalities for strongly η-convex stochastic processes [156]. Theorem 7.5.2. Suppose that ξ : [c, d] × Ω → R is an strongly η−convex stochastic process with modulus µ(.) > 0, such that η is bounded above on ξ[c, d] × ξ[c, d], then the following inequalities hold almost everywhere:    d Mη µ(.) 1 c+d ,. − + (d − c)2 ≤ ξ(x, .)dx ξ 2 2 12 d−c c µ(.) ξ(c, .) + ξ(d, .) 1 + (η(ξ(c, .), ξ(d, .)) + η(ξ(d, .), ξ(c, .))) − (d − c)2 ≤ 2 4 6 ξ(c, .) + ξ(d, .) µ(.) + Mη − (d − c)2 , ≤ 2 6 where Mη is an upper bound of η.

218

Integral Inequalities and Generalized Convexity

Proof Since ξ is strongly η-convex stochastic process, therefore         c+d 1 c + d − δ(d − c) 1 c + d + δ(d − c) ξ ,. = ξ + ,. 2 2 2 2 2   c + d + δ(d − c) ≤ξ ,. 2      µ(.) 2 1 c + d − δ(d − c) c + d + δ(d − c) ,. ,ξ ,. − δ (d − c)2 + η ξ 2 2 2 4   Mη µ(.) 2 c + d + δ(d − c) ,. + − δ (d − c)2 . ≤ξ 2 2 4 This implies,     c+d Mη µ(.) 2 c + d + δ(d − c) 2 ξ ,. − + δ (d − c) ≤ ξ ,. . 2 2 4 2 Similarly,     c+d Mη µ(.) 2 c + d − δ(d − c) ξ ,. − + δ (d − c)2 ≤ ξ ,. . 2 2 4 2 By using the change of variable technique, we have    d  d (c+d)/2 1 1 ξ(x, .)dx = ξ(x, .)dx + ξ(x, .)dx d−c c d−c c (c+d)/2       1 1 c + d − δ(d − c) c + d + δ(d − c) 1 1 , . dδ + , . dδ = ξ ξ 2 0 2 2 0 2      1 Mη µ(.) 2 c+d ≥ ξ ,. − + δ (d − c)2 dδ 2 2 4 0   c+d Mη µ(.) =ξ ,. − + (d − c)2 . (7.19) 2 2 12 We now prove the right hand side of the theorem. Since ξ is strongly η-convex stochastic process with modulus µ(.) > 0, we get ξ(δc + (1 − δ)d, .) ≤ ξ(d, .) + δη(ξ(c, .), ξ(d, .)) − µ(.)δ(1 − δ)(d − c)2 . Integrating above inequality with respect to δ on both sides from 0 to 1, we have  1 ξ(δc + (1 − δ)d, .)dδ 0

≤



0

1

(ξ(d, .) + δη(ξ(c, .), ξ(d, .)) − µ(.)δ(1 − δ)(d − c)2 )dδ.

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 219 This implies, 1 d−c



d

µ(.) 1 (d − c)2 = P. ξ(x, .)dx ≤ ξ(d, .) + η(ξ(c, .), ξ(d, .)) − 2 6

d

µ(.) 1 (d − c)2 = Q. ξ(x, .)dx ≤ ξ(c, .) + η(ξ(d, .), ξ(c, .)) − 2 6

c

Similarly, 1 d−c



c

Therefore, 1 d−c



d c

ξ(x, .)dx ≤ M in{P, Q}

ξ(c, .) + ξ(d, .) 1 + (η(ξ(c, .), ξ(d, .)) + η(ξ(d, .), ξ(c, .))) 2 4 µ(.) 2 − (d − c) 6 ξ(c, .) + ξ(d, .) µ(.) + Mη − (d − c)2 . ≤ (7.20) 2 6 ≤

From (7.19) and (7.20), we have ξ



  d Mη µ(.) 1 c+d ,. − + (d − c)2 ≤ ξ(x, .)dx 2 2 12 d−c c ξ(c, .) + ξ(d, .) 1 µ(.) ≤ + (η(ξ(c, .), ξ(d, .)) + η(ξ(d, .), ξ(c, .))) − (d − c)2 2 4 6 ξ(c, .) + ξ(d, .) µ(.) + Mη − (d − c)2 . ≤ (7.21) 2 6

Remark 7.5.3. When µ(.) = 0, then above theorem reduces to Theorem 7.2.7. If we consider η(ξ(x, .), ξ(y, .)) = ξ(x, .) − ξ(y, .) and µ(.) = 0, then above theorem reduces to the classical Hermite–Hadamard inequality for convex stochastic process [87]. Theorem 7.5.3. If a stochastic process ξ : K ×Ω → R be an strongly η-convex with modulus µ(.) > 0 and integrable on K × Ω, we have 

d

1 ξ(x, .)ξ(c + d − x, .)dx ≤ ξ(c, .)ξ(d, .) + (ξ(c, .)η(ξ(c, .), ξ(d, .)) 2 c 1 + ξ(d, .)η(ξ(d, .), ξ(c, .))) + η(ξ(c, .), ξ(d, .))η(ξ(d, .), ξ(c, .)) 3 µ(.) µ(.) (d − c)2 (ξ(c, .) + ξ(d, .)) − (d − c)2 (η(ξ(c, .), ξ(d, .)) − 6 12 µ2 (.) + η(ξ(d, .), ξ(c, .))) + (d − c)4 (a.e.). 30

1 d−c

220

Integral Inequalities and Generalized Convexity

Proof Since ξ is strongly η-convex stochastic process, therefore ξ(δc + (1 − δ)d, .) ≤ ξ(d, .) + δη(ξ(c, .), ξ(d, .)) − µ(.)δ(1 − δ)(d − c)2 (7.22) and ξ(δd + (1 − δ)c, .) ≤ ξ(c, .) + δη(ξ(d, .), ξ(c, .)) − µ(.)δ(1 − δ)(d − c)2 . (7.23) From (7.22) and (7.23), we obtain ξ(δc + (1 − δ)d, .)ξ(δd + (1 − δ)c, .) ≤ ξ(c, .)ξ(d, .) + δ(ξ(c, .)η(ξ(c, .), ξ(d, .)) + ξ(d, .)η(ξ(d, .), ξ(c, .))) + δ 2 η(ξ(c, .), ξ(d, .))η(ξ(d, .), ξ(c, .)) − µ(.)δ(1 − δ)(d − c)2 (ξ(c, .) + ξ(d, .)) − µ(.)δ 2 (1 − δ)(d − c)2 (η(ξ(c, .), ξ(d, .)) + η(ξ(d, .), ξ(c, .))) + µ2 (.)δ 2 (1 − δ)2 (d − c)4 . Integrating above inequality from 0 to 1 on both sides with respect to δ, we have  1 ξ(δc + (1 − δ)d, .)ξ(δd + (1 − δ)c, .)dδ ≤ ξ(c, .)ξ(d, .) 0

1 + (ξ(c, .)η(ξ(c, .), ξ(d, .)) + ξ(d, .)η(ξ(d, .), ξ(c, .))) 2 1 µ(.) + η(ξ(c, .), ξ(d, .))η(ξ(d, .), ξ(c, .)) − (d − c)2 (ξ(c, .) 3 6 + ξ(d, .)) −

µ(.) µ2 (.) (d − c)2 (η(ξ(c, .), ξ(d, .)) + η(ξ(d, .), ξ(c, .))) + (d − c)4 . 12 30

This implies, 1 d−c



d c

1 ξ(x, .)ξ(c + d − x, .)dx ≤ ξ(c, .)ξ(d, .) + (ξ(c, .)η(ξ(c, .), ξ(d, .)) 2

1 + ξ(d, .)η(ξ(d, .), ξ(c, .))) + η(ξ(c, .), ξ(d, .))η(ξ(d, .), ξ(c, .)) 3 −

µ(.) µ(.) (d − c)2 (ξ(c, .) + ξ(d, .)) − (d − c)2 (η(ξ(c, .), ξ(d, .)) 6 12

+ η(ξ(d, .), ξ(c, .))) +

µ2 (.) (d − c)4 . 30

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 221 Remark 7.5.4. When ξ is strongly log-convex stochastic process, then above theorem reduces to Theorem 3 of [166], i.e. 1 d−c



d c

ξ(x, .)ξ(c + d − x, .)dx ≤ ξ(c, .)ξ(d, .) +

µ2 (.) (d − c)4 30

4µ(.)(d − c)2 [A(ξ(c, .), ξ(d, .)) + L(ξ(c, .), ξ(d, .))] ln[ξ(c, .) − ξ(d, .)]2 2[A(ξ(c, .), ξ(d, .))]2 + [G(ξ(c, .), ξ(d, .))]2 ≤ 3 µ(.)A(ξ(c, .), ξ(d, .))(d − c)2 µ2 (.) − + (d − c)4 . 3 30 −

Theorem 7.5.4. Let ξ : K × Ω → R be a mean square stochastic process such that ξ  is mean square integrable on [c, d], where c, d ∈ K with c < d. If |ξ  | is an strongly η-convex stochastic process with modulus µ(.) > 0 on K and |ξ  (t, .)| ≤ M for every t, then    d   1   ξ(x, .)dx ξ(t, .) −   d−c c 1 M ((t − c)2 + (d − t)2 ) + [(t − c)2 η(|ξ  (t, .)|, |ξ  (c, .)|) 2(d − c) 3(d − c) µ(.) + (d − t)2 η(|ξ  (t, .)|, |ξ  (d, .)|)] − ((t − c)4 + (d − t)4 ) (a.e.). 12(d − c)

≤

Proof Recall Lemma 7.2.1:  d  1 (t − c)2 1  ξ(t, .) − ξ(x, .)dx = yξ (yt + (1 − y)c, .)dy d−c c d−c 0  (d − t)2 1  − yξ (yt + (1 − y)d, .)dy, (a.e), for each t ∈ [c, d]. d−c 0 Since |ξ  | is strongly η-convex stochastic process, therefore    d   1   ξ(x, .)dx ξ(t, .) −   d−c c  (t − c)2 1 ≤ y[|ξ  (c, .)| + yη(|ξ  (t, .)|, |ξ  (c, .)|) − µ(.)y(1 − y)(c − t)2 ]dy d−c 0  (d − t)2 1 + y[|ξ  (d, .)| + yη(|ξ  (t, .)|, |ξ  (d, .)|) − µ(.)y(1 − y)(d − t)2 ]dy d−c 0   η(|ξ  (t, .)|, |ξ  (c, .)|) µ(.) (t − c)2 M 2 + − (c − t) ≤ d−c 2 3 12   (d − t)2 M η(|ξ  (t, .)|, |ξ  (d, .)|) µ(.) 2 + + − (d − t) d−c 2 3 12

222

Integral Inequalities and Generalized Convexity 1 M ((t − c)2 + (d − t)2 ) + [(t − c)2 η(|ξ  (t, .)|, |ξ  (c, .)|) 2(d − c) 3(d − c) µ(.) ((t − c)4 + (d − t)4 ). + (d − t)2 η(|ξ  (t, .)|, |ξ  (d, .)|)] − 12(d − c)

=

Remark 7.5.5. When µ(.) = 0, then above theorem reduces to Theorem 4.2 of [69], i.e.    d   1   ξ(x, .)dx ξ(t, .) −   d−c c ≤

1 M ((t − c)2 + (d − t)2 ) + [(t − c)2 η(|ξ  (t, .)|, |ξ  (c, .)|) 2(d − c) 3(d − c)

+ (d − t)2 η(|ξ  (t, .)|, |ξ  (d, .)|)] (a.e.).

Theorem 7.5.5. Let ξ : K ×Ω → R be a mean square differentiable stochastic process on K 0 and ξ  be a mean square integrable on [c, d], where c, d ∈ K, c < d. If |ξ  | is an strongly η-convex stochastic process on [c, d], then we have almost everywhere:    d  ξ(c, .) + ξ(d, .)  1   − ξ(x, .)dx    2 d−c c   d−c 1 µ(.) |ξ  (d, .)| + η(|ξ  (c, .)|, |ξ  (d, .)|) − (d − c)2 . ≤ 4 2 8 Proof From Lemma 7.2.2, we have    d  ξ(c, .) + ξ(d, .)  d−c 1 1   − ξ(x, .)dx ≤ |1 − 2δ||ξ  (δc + (1 − δ)d, .)|dδ.    2 d−c c 2 0

Using the definition of strong η-convex stochastic process in above inequality, we have    d  ξ(c, .) + ξ(d, .)  1   − ξ(x, .)dx    2 d−c c  1 d−c ≤ |1 − 2δ|(|ξ  (d, .)| + δη(|ξ  (c, .)|, |ξ  (d, .)|) 2 0 − µ(.)δ(1 − δ)(d − c)2 )dδ  1  1 d−c  = |1 − 2δ||ξ (d, .)|dδ + δ|1 − 2δ|η(|ξ  (c, .)|, |ξ  (d, .)|)dδ 2 0 0   1 2 − µ(.)(d − c) δ(1 − δ)|1 − 2δ|dδ 0

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 223   d−c 1  1 µ(.) |ξ (d, .)| + η(|ξ  (c, .)|, |ξ  (d, .)|) − (d − c)2 = 2 2 4 16   d−c 1 µ(.)    2 = |ξ (d, .)| + η(|ξ (c, .)|, |ξ (d, .)|) − (d − c) . 4 2 8 Corollary 7.5.1. When µ(.) = 0 in above theorem, then we obtain the following inequality for η-convex stochastic processes:    d   ξ(c, .) + ξ(d, .) 1   − ξ(x, .)dx    2 d−c c ≤

d−c 4



1 |ξ  (d, .)| + η(|ξ  (c, .)|, |ξ  (d, .)|) 2



(a.e.).

Remark 7.5.6. When µ(.) = 0 and η(|ξ  (c, .)|, |ξ  (d, .)|) = |ξ  (c, .)| − |ξ  (d, .)|, then above theorem reduces to Corollary 5.3 of Fu et al. [51], i.e.    d  d−c  ξ(c, .) + ξ(d, .) 1   − A(|ξ  (c, .)|, |ξ  (d, .)|) (a.e.). ξ(x, .)dx ≤    2 d−c c 4

Theorem 7.5.6. Let ξ : K ×Ω → R be a mean square differentiable stochastic process on K 0 with q > 1, p1 + 1q = 1 and assume that ξ  be a mean square integrable on [c, d], where c, d ∈ K, c < d. If |ξ  |q is an strongly η-convex stochastic process on [c, d], then we have almost everywhere:    d  d − c  1 1/p  ξ(c, .) + ξ(d, .) 1   − ξ(x, .)dx =    2 d−c c 2 p+1 

1 µ(.) (d − c)2 × |ξ (d, .)| + η(|ξ  (c, .)|q , |ξ  (d, .)|q ) − 2 6 

q

Proof From Lemma 7.2.2, we have    d  ξ(c, .) + ξ(d, .)  1   − ξ(x, .)dx    2 d−c c ≤

d−c 2



1

0

|1 − 2δ||ξ  (δc + (1 − δ)d, .)|dδ.

1/q

.

224

Integral Inequalities and Generalized Convexity

Using H¨ older’s inequality and the definition of strong η-convex stochastic process in above inequality, we obtain    d   ξ(c, .) + ξ(d, .) 1   − ξ(u, .)du    2 d−c c  1 1/p  1 1/q d−c |1 − 2δ|p dδ |ξ  (δc + (1 − δ)d, .)|q dδ ≤ 2 0 0  1/p  1 d−c 1 ≤ (|ξ  (d, .)|q + δη(|ξ  (c, .)|q , |ξ  (d, .)|q ) 2 p+1 0 1/q − µ(.)δ(1 − δ)(d − c)2 )dδ  1/p d−c 1 = 2 p+1 1/q  1 µ(.) (d − c)2 × |ξ  (d, .)|q + η(|ξ  (c, .)|q , |ξ  (d, .)|q ) − . 2 6 Corollary 7.5.2. When µ(.) = 0 in above theorem, then we obtain the following inequality for η-convex stochastic processes:    d  ξ(c, .) + ξ(d, .)  1   − ξ(x, .)dx    2 d−c c  1/p  1/q 1 1 d−c (a.e.). |ξ  (d, .)|q + η(|ξ  (c, .)|q , |ξ  (d, .)|q ) ≤ 2 p+1 2 Remark 7.5.7. When µ(.) = 0 and η(|ξ  (c, .)|q , |ξ  (d, .)|q ) = |ξ  (c, .)|q − |ξ  (d, .)|q , then above theorem reduces to Corollary 5.5 of Fu et al. [51], i.e.    d   ξ(c, .) + ξ(d, .) 1   − ξ(x, .)dx    2 d−c c  1/p 1 d−c 1 A q (|ξ  (c, .)|q , |ξ  (d, .)|q ) (a.e.). ≤ 2 p+1 Theorem 7.5.7. Let ξ : K ×Ω → R be a mean square differentiable stochastic process on K 0 with q ≥ 1, and assume that ξ  be a mean square integrable on [c, d], where c, d ∈ K, c < d. If |ξ  |q is an strongly η-convex stochastic process on [c, d], then we have almost everywhere:    d  ξ(c, .) + ξ(d, .)  1   − ξ(x, .)dx    2 d−c c  1/q 1 µ(.) d−c |ξ  (d, .)|q + η(|ξ  (c, .)|q , |ξ  (d, .)|q ) − (d − c)2 . ≤ 4 2 8

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 225 Proof For q = 1, we use the estimates from the proof of Theorem 7.5.5. Now we prove result for q > 1. From Lemma 7.2.2, we have    d  ξ(c, .) + ξ(d, .)  1   − ξ(x, .)dx    2 d−c c  1 d−c ≤ |1 − 2δ||ξ  (δc + (1 − δ)d, .)|dδ. 2 0 Using H¨ older’s inequality and the definition of strong η-convex stochastic process in above inequality, we obtain    d  ξ(c, .) + ξ(d, .)  1   − ξ(x, .)dx    2 d−c c  1 1− q1  1 1/q d−c  q |1 − 2δ|dδ |1 − 2δ||ξ (δc + (1 − δ)d, .)| dδ ≤ 2 0 0  1− q1  1 d−c 1 ≤ |1 − 2δ|(|ξ  (d, .)|q + δη(|ξ  (c, .)|q , |ξ  (d, .)|q ) 2 2 0 1/q −µ(.)δ(1 − δ)(d − c)2 )dδ 1/q  1− q1  d−c 1 1 µ(.) 1  |ξ (d, .)|q + η(|ξ  (c, .)|q , |ξ  (d, .)|q ) − (d − c)2 = 2 2 2 4 16  1/q d−c 1 µ(.)  q  q  q 2 |ξ (d, .)| + η(|ξ (c, .)| , |ξ (d, .)| ) − (d − c) = . 4 2 8 Corollary 7.5.3. When µ(.) = 0 in above theorem, then we obtain the following inequality for η-convex stochastic processes:    d  ξ(c, .) + ξ(d, .)  1   − ξ(x, .)dx    2 d−c c  1/q 1 d−c  q  q  q |ξ (d, .)| + η(|ξ (c, .)| , |ξ (d, .)| ) (a.e.). ≤ 4 2 Remark 7.5.8. When µ(.) = 0 and η(|ξ  (c, .)|q , |ξ  (d, .)|q ) = |ξ  (c, .)|q − |ξ  (d, .)|q , then above theorem reduces to Corollary 5.8 of Fu et al. [51], i.e.    d   ξ(c, .) + ξ(d, .) 1   − ξ(x, .)dx    2 d−c c ≤

d − c q1  A (|ξ (c, .)|q , |ξ  (d, .)|q ) (a.e.). 4

Theorem 7.5.8. Let ξ : K ×Ω → R be a mean square differentiable stochastic process on K 0 with q > 1, p1 + 1q = 1 and assume that ξ  be a mean square

226

Integral Inequalities and Generalized Convexity

integrable on [c, d], where c, d ∈ K, c < d. If |ξ  |q is an strongly η-convex stochastic process on [c, d], then we have almost everywhere:    d  ξ(c, .) + ξ(d, .)  1   − ξ(x, .)dx    2 d−c c 1  p 1 d−c ≤ 2 2(p + 1)  1/q 1  1 µ(.) q  q  q 2 |ξ (d, .)| + η(|ξ (c, .)| , |ξ (d, .)| ) − (d − c) × 2 6 12 1/q   1 µ(.) 1  q  q  q 2 . (|ξ (d, .)| + η(|ξ (c, .)| , |ξ (d, .)| ) − (d − c) + 2 3 12 Proof From Lemma 7.2.2, we have    d  ξ(c, .) + ξ(d, .)  1   − ξ(x, .)dx    2 d−c c  1 d−c ≤ |1 − 2δ||ξ  (δc + (1 − δ)d, .)|dδ. 2 0 Using H¨ older’s inequality and the definition of strong η-convex stochastic process in above inequality, we get    d  ξ(c, .) + ξ(d, .)  1   ξ(x, .)dx −   2 d−c c  p1  1 1/q  1 d−c ≤ (1 − δ)|1 − 2δ|p dδ (1 − δ)|ξ  (δc + (1 − δ)d, .)|q dδ 2 0 0  p1  1 1/q  1 d−c + δ|1 − 2δ|p dδ δ|ξ  (δc + (1 − δ)d, .)|q dδ 2 0 0  1 p d−c 1 ≤ 2 2(p + 1)  1

× +

0



1 0

(1 − δ)(|ξ  (d, .)|q + δη(|ξ  (c, .)|q , |ξ  (d, .)|q ) − µ(.)δ(1 − δ)(d − c)2 )dδ

δ(|ξ  (d, .)|q + δη(|ξ  (c, .)|q , |ξ  (d, .)|q ) − µ(.)δ(1 − δ)(d − c)2 )dδ

1/q 

1/q

1/q   1  p µ(.) d−c 1 1  1 = |ξ (d, .)|q + η(|ξ  (c, .)|q , |ξ  (d, .)|q ) − (d − c)2 2 2(p + 1) 2 6 12  1/q  µ(.) 1  1 + . (|ξ (d, .)|q + η(|ξ  (c, .)|q , |ξ  (d, .)|q ) − (d − c)2 2 3 12

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 227 Corollary 7.5.4. When µ(.) = 0 in above theorem, then we obtain the following inequality for η-convex stochastic processes:    d   ξ(c, .) + ξ(d, .) 1   − ξ(x, .)dx    2 d−c c    p1  1/q 1 1 1  d−c |ξ (d, .)|q + η(|ξ  (c, .)|q , |ξ  (d, .)|q ) ≤ 2 2(p + 1) 2 6  1/q  1  1 (|ξ (d, .)|q + η(|ξ  (c, .)|q , |ξ  (d, .)|q ) + (a.e.). 2 3 Remark 7.5.9. When µ(.) = 0 and η(|ξ  (c, .)|q , |ξ  (d, .)|q ) = |ξ  (c, .)|q − |ξ  (d, .)|q , then above theorem reduces to Corollary 5.10 of Fu et al. [51], i.e.    d  d − c  1  p1  ξ(c, .) + ξ(d, .) 1   − ξ(x, .)dx ≤    2 d−c c 4 (p + 1)   1/q   1/q |ξ  (c, .)|q + 2|ξ  (d, .)|q 2|ξ  (c, .)|q + |ξ  (d, .)|q × + (a.e.). 3 3 Theorem 7.5.9. Let ξ : K ×Ω → R be a mean square differentiable stochastic process on K 0 with q ≥ 1, and assume that ξ  be a mean square integrable on [c, d], where c, d ∈ K, c < d. If |ξ  |q is an strongly η-convex stochastic process on [c, d], then we have almost everywhere:    d  ξ(c, .) + ξ(d, .)  1   − ξ(x, .)dx    2 d−c c  1/q d−c 1 µ(.) (d − c)2 |ξ  (d, .)|q + η(|ξ  (c, .)|q , |ξ  (d, .)|q ) − ≤ 8 4 8  1/q  3 µ(.)  q  q  q 2 (d − c) × |ξ (d, .)| + η(|ξ (c, .)| , |ξ (d, .)| ) − . 4 8 Proof For q = 1, we use the estimates from the proof of Theorem 7.5.5. Now we prove result for q > 1. From Lemma 7.2.2, we have    d  ξ(c, .) + ξ(d, .)  1   − ξ(x, .)dx    2 d−c c  1 d−c ≤ |1 − 2δ||ξ  (δc + (1 − δ)d, .)|dδ. 2 0

228

Integral Inequalities and Generalized Convexity

Using improved power-mean integral inequality and the definition of strong η-convex stochastic process in above inequality, we get    d   ξ(c, .) + ξ(d, .) 1  ξ(x, .)dx −  2 d−c c 1− q1  1 1/q  1 d−c ≤ (1 − δ)|1 − 2δ|dδ (1 − δ)|1 − 2δ||ξ  (δc + (1 − δ)d, .)|q dδ 2 0 0 1− q1  1 1/q  1 d−c + δ|1 − 2δ|dδ δ|1 − 2δ||ξ  (δc + (1 − δ)d, .)|q dδ 2 0 0  1− 1  1 q d−c 1 ≤ (1 − δ)|1 − 2δ|(|ξ  (d, .)|q + δη(|ξ  (c, .)|q , |ξ  (d, .)|q ) 2 4 0  1 1/q + δ|1 − 2δ|(|ξ  (d, .)|q + δη(|ξ  (c, .)|q , |ξ  (d, .)|q ) − µ(.)δ(1 − δ)(d − c)2 )dδ 0 1/q  − µ(.)δ(1 − δ)(d − c)2 )dδ 1/q  1− 1  q µ(.) d−c 1 1 1  = |ξ (d, .)|q + η(|ξ  (c, .)|q , |ξ  (d, .)|q ) − (d − c)2 2 4 4 16 32  1/q  µ(.) 1  3 + |ξ (d, .)|q + η(|ξ  (c, .)|q , |ξ  (d, .)|q ) − (d − c)2 4 16 32  1/q µ(.) d−c 1  q  q  q 2 = |ξ (d, .)| + η(|ξ (c, .)| , |ξ (d, .)| ) − (d − c) 8 4 8  1/q  µ(.) 3  q  q  q 2 + |ξ (d, .)| + η(|ξ (c, .)| , |ξ (d, .)| ) − . (d − c) 4 8

Corollary 7.5.5. When µ(.) = 0 in above theorem, then we obtain the following inequality for η-convex stochastic processes:    d  ξ(c, .) + ξ(d, .)  1   − ξ(x, .)dx    2 d−c c  1/q d−c 1  q  q  q |ξ (d, .)| + η(|ξ (c, .)| , |ξ (d, .)| ) ≤ 8 4  1/q  3  q  q  q (a.e.). + |ξ (d, .)| + η(|ξ (c, .)| , |ξ (d, .)| ) 4

Some Inequalities for Multidimensional General h-Harmonic Preinvex..... 229 Remark 7.5.10. When µ(.) = 0 and η(|ξ  (c, .)|q , |ξ  (d, .)|q ) = |ξ  (c, .)|q − |ξ  (d, .)|q , then above theorem reduces to Corollary 5.13 of Fu et al. [51], i.e.    d  ξ(c, .) + ξ(d, .)  1   − ξ(x, .)dx    2 d−c c  1/q   1/q  3|ξ  (d, .)|q |ξ  (d, .)|q d−c |ξ  (c, .)|q 3|ξ (c, .)|q + + ≤ + 8 4 4 4 4 (a.e.).

Chapter 8 Applications

8.1

Hermite–Hadamard Inequality

The Hermite–Hadamard inequality [56] is the first fundamental result for convex functions defined on a interval of real numbers with a natural geometrical interpretation. Hermite–Hadamard inequality is now itself a special domain of the theory of inequalities with many powerful results and a large number of applications in numerical integration, probability theory and statistics, information theory and integral operator theory. The following examples show the importance of Hermite–Hadamard inequality in Calculus. Example 8.1.1. For ξ(x) =

1 2

 0 and c = 0, d = x in Theorem 1.3.1, 2

x log(1 + x) < x − 2(1+x) . In particular,   1 1 1 + < log(n + 1) − log n < 2 n n+1

Hermite [59] noticed that x − 1 n+

1 1+x , x x2 x+2