Nevanlinna Theory and Complex Differential Equations 9783110863147, 9783110134223

Table of contents :
Introduction
Chapter 1. Results from function theory
Chapter 2. Nevanlinna theory of meromorphic functions
Chapter 3. Wiman-Valiron theory
Chapter 4. Linear differential equations: basic results
Chapter 5. Linear differential equations: zero distribution in the second order case
Chapter 6. Complex differential equations and the Schwarzian derivative
Chapter 7. Higher order linear differential equations
Chapter 8. Non-homogeneous linear differential equations
Chapter 9. Basic non-linear differential equations
Chapter 10. The Malmquist-Yosida-Steinmetz type theorems
Chapter 11. First order algebraic differential equations
Chapter 12. Second order algebraic differential equations
Chapter 13. Algebraic differential equations of arbitrary order
Chapter 14. Algebraic differential equations and differential fields
Bibliography
Index

Citation preview

de Gruyter Studies in Mathematics 15 Editors: Heinz Bauer · Jerry L. Kazdan · Eduard Zehnder

de Gruyter Studies in Mathematics 1 Riemannian Geometry, Wilhelm Klingenberg 2 Semimartingales, Michel Metivier 3 Holomorphic Functions of Several Variables, Ludger Kaup and Burchard Kaup 4 Spaces of Measures, Corneliu Constantinescu 5 Knots, Gerhard Burde and Heiner Zieschang 6 Ergodic Theorems, Ulrich Krengel 7 Mathematical Theory of Statistics, Helmut Strasser 8 Transformation Groups, Tammo torn Dieck 9 Gibbs Measures and Phase Transitions, Hans-Otto Georgii 10 Analyticity in Infinite Dimensional Spaces, Michel Herve 11 Elementary Geometry in Hyperbolic Space, Werner Fenchel 12 Transcendental Numbers, Andrei B. Shidlovskii 13 Ordinary Differential Equations, Herbert Amann 14 Dirichlet Forms and Analysis on Wiener Space, Nicolas Bouleau and Francis Hirsch

lipo Laine

Nevanlinna Theory and Complex Differential Equations

w

Walter de Gruyter G Berlin · New York 1993 DE

Author lipo Laine Department of Mathematics University of Joensuu P.O. Box 111 SF-80101 Joensuu Finland

Series Editors Heinz Bauer Mathematisches Institut der Universität Bismarckstrasse 1 Ά D-8520 Erlangen, FRG

Jerry L. Kazdan Department of Mathematics University of Pennsylvania 209 South 33rd Street Philadelphia, PA 19104-6395, USA

Eduard Zehnder ETH-Zentrum/Mathematik Rämistrasse 101 CH-8092 Zürich Switzerland

1991 Mathematics Subject Classification: 30-02; 34-02; 30D35, 34A20

©

Printed on acid-free paper which falls within the guidelines of the A N S I to ensure permanence and durability.

Library of Congress Cataloging-in-Publication Data Laine, lipo. Nevanlinna theory and complex differential equations / lipo Laine. p. cm. — (De Gruyter studies in mathematics ; 15) Includes bibliographical references and index. ISBN 3-11-013422-5 (alk. paper) 1. Nevanlinna theory. 2. Differential equations. 3. Functions of complex variables. I. Title. II. Series. QA331.L24 1993 515'.35—dc20 92-35852 CIP

Die Deutsche Bibliothek — Cataloging-in-Publication Data Laine, Προ: Nevanlinna theory and complex differential equations / lipo Laine. — Berlin ; New York : de Gruyter, 1993 (De Gruyter studies in mathematics ; 15) ISBN 3-11-013422-5 NE: GT

© Copyright 1992 by Walter de Gruyter & Co., D-1000 Berlin 30. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Disk conversion: Danny Lee Lewis, Berlin. Printing: Gerike GmbH, Berlin. Binding: Dieter Mikolai, Berlin. Cover design: Rudolf Hübler, Berlin.

Preface

This book arose from a relatively long process. The idea first appeared many years ago, due to my longstanding research collaboration with Professor Steven Bank (Urbana). While visiting Urbana in 1987,1 prepared the first draft, which appeared later on as a survey article in Finnish. The second draft was used for my lectures at the University of Erlangen in 1989. Several parts of the manuscript also served as a basis for a large number of graduate seminars at the University of Joensuu. Finally, the finishing touch was made during the Spring School in Potential Theory, organized by the University of Prague, in the inspiring scenery of the Krkonose Mountains in April 1992. From the many colleagues who owe my gratitude, Professor Steven Bank (Urbana) had the most essential influence on the subjectmatter of this book, due to our numerous discussions during almost twenty years. He also commented several parts of the manuscript. Nevertheless, the responsibility for any errors or defects in the book is mine. Professor Heinz Bauer (Erlangen) encouraged me to complete the writing process. He also proposed that the book could be accepted in the Walter de Gruyter series Studies in Mathematics. Concerning the staff at the Department of Mathematics, University of Joensuu, I am pleased to mention in public Kari Katajamäki and Liisa Kinnunen who helped me by checking a multitude of details and by commenting several versions of various chapters. Their efforts can be found in many passages throughout my exposition. My research secretary, Riitta Heiskanen, converted my drafts into a readable form with an admirable skill. Academy of Finland, University of Joensuu and University of Erlangen have been the institutions which provided me with financial support. With a probability which is very close to one, this book would have never appeared without this support. Last, but definitely not least, my warm thanks are due to my family. Their patience with the seemingly endless process of writing has been simply astonishing. I hope they have not been persistent in vain. Joensuu, April 1992

lipo Laine

Contents

Introduction

1

Chapter 1 Results from function theory

5

Chapter 2 Nevanlinna theory of meromorphic functions

18

Chapter 3 Wiman-Valiron theory

50

Chapter 4 Linear differential equations: basic results

53

Chapter 5 Linear differential equations: zero distribution in the second order case

.

74

Chapter 6 Complex differential equations and the Schwarzian derivative

110

Chapter 7 Higher order linear differential equations

127

Chapter 8 Non-homogeneous linear differential equations

144

Chapter 9 Basic non-linear differential equations

165

Chapter 10 The Malmquist Yosida-Steinmetz type theorems

192

Chapter 11 First order algebraic differential equations

221

Chapter 12 Second order algebraic differential equations

248

viii

Contents

Chapter 13 Algebraic differential equations of arbitrary order

257

Chapter 14 Algebraic differential equations and differential

fields

285

Bibliography

311

Index

339

Introduction

Differential equations in the complex domain is an area of mathematics admitting several ways of approach. The local theory is perhaps the most investigated of these approaches. Its basic results, say the local existence and uniqueness theorem of solutions, singularity theory etc., can be found in a large number of text-books of differential equations. Our focus of interest is different. The basic existence and uniqueness theorem and the basic linear structure of solutions of linear differential equations are the only results from the local theory we assume the reader is familiar with in advance. The global theory, where our interest lies, can also be studied in many different ways. For instance, one may consider it from the algebraic point of view, see e.g. Matsuda [1], from the differential equations point of view, see e.g. Jurkat [1], or one may consider complex differential equations from the direction of function theory, which is the approach of this exposition. Precisely, our aim has been to show how the Nevanlinna theory may be applied to get insight into the properties of solutions of complex differential equations. The first such applications were made, to the best of our knowledge, by F. Nevanlinna [1] in 1929, who considered the differential equation f"+ A{z)f = 0 in the case of a polynomial A(z) in connection of a study of meromorphic functions with maximal deficiency sum, by R. Nevanlinna [1], who considered the same equation in connection of covering surfaces with finitely many branch points and by K. Yosida [l]-[5] who proved the celebrated Malmquist theorem via the Nevanlinna theory in [1]. The first one who made systematic studies in the applications of Nevanlinna theory into complex differential equations was H. Wittich beginning from 1942. From the early contributions in this area we still add the article by A. Gol'dberg [ 1 ] which is perhaps the most important paper treating general algebraic differential equations before seventies. Now, apart from a few other earlier contributions, one had to wait up to the end of sixties before the global theory of complex differential equations, in connection with Nevanlinna theory, became more popular. During the last two decades several active groups of mathematicians in different countries have played a remarkable role. However, up to now, only a few survey articles (see Nikolaus [6], Eremenko [31, Mues [7]) and a relatively small number of books (see e.g. Bieberbach [1], Wittich [9], Herold [1], Hille [3], Petrenko [1], Jank and Volkmann [3], He and Xiao [5]) exist where at least a substantial part has been devoted to complex differential equations. Despite of the many advantages of these surveys and books, some of their limitations perhaps justify to offer a new reference. Especially, concerning most of the above books, their main emphasis is something else than the title of this book. The best source hitherto is perhaps the book by Jank and Volkmann [3], p. 163-241; nevertheless, the major part of [3] is

2

Introduction

devoted to the Nevanlinna theory itself. To avoid unnecessary repetition, we have omitted here some material and some technically complicated proofs which can be found in the book of Jank and Volkmann. The aim of this book is twofold. First of all, we have tried to give a concise treatment of Nevanlinna theory applications into the global theory of complex differential equations, beginning from the classical results and ending, at least for the major part, up to current research trends. Reading of this book certainly becomes easier, if some familiarity with the Nevanlinna theory is preassumed. As mentioned above, the prerequisites from the theory of differential equations are practically nonexisting, restricting themselves to the usual undergraduate material. Our second aim lies in the hope that an interested graduate student could find here an easy access to this special area of differential equations. The most important mathematical background a graduate student needs is a reasonable working knowledge of function theory, say e.g. on the level of Veech [1]. We also hope that the material of this book could be useful for graduate level lecture courses and seminars. The material in this book has been organized as follows. The first three chapters contain some background material, mostly from function theory. Specially, Chapter 2 collects the relevant parts of the Nevanlinna theory, from the point of view of applications into differential equations. If a proof has been omitted here, an exact reference has been pointed out. One of the key results in Chapter 2 is the famous Clunie lemma, see Section 2.4, presented here in all details, including several variants and related results. Chapter 3 contains the central results from the Wiman-Valiron theory, sometimes more powerful than the Nevanlinna theory itself, to estimate the order of growth of an entire function. With one exception only, proofs have been omitted in this short chapter. This is due to the fact that Jank and Volkmann [3], Chapters 4 and 21, gives an excellent exposition the reader will be advised to consult. The next five chapters are devoted to linear differential equations. Chapters 4 through 6 form the elementary part, consisting of basic applications of the Nevanlinna theory, mostly into the second order homogeneous linear differential equations in the complex plane. Despite of their elementary nature, most of what can be found in Chapter 5 and Chapter 6 originates from the last decade. Concerning higher order linear differential equations and non-homogeneous differential equations, it may be somewhat surprising that Nevanlinna theory applications are still fairly few. However, some recent investigations, see Frank and Hellerstein [1], Bank and Langley [3] and Langley [7] show that surprisingly much similarity with the second order, resp. homogeneous, case appears. The problem is that the methods of proof tend to be rather complicated. Therefore, Chapter 7 and Chapter 8 don't try to be complete. On the contrary, we have tried to extract from the existing very few original sources those parts where the proofs remain reasonably simple. The last six chapters consider non-linear algebraic differential equations, making some occasional returns back to the linear theory. Chapter 9 gives a short

Introduction

3

excursion into the simple non-linear equations, consisting of the Riccati equation, some of the Painleve equations, and the Schwarzian differential equation, while Chapter 10 gives a justification to their special status via the nowadays classical Malmquist argument. For the Riccati type differential equations, this argument gives a complete classification today, while the situation remains incomplete for the Painleve type. Chapters 11 through 13 then present Nevanlinna theory applications into the global theory of algebraic differential equations in general. The final Chapter 14 is of a bit different nature, written to show how Nevanlinna theory and abstract algebraic reasoning together may produce interesting results in the area of complex differential equations. We feel that Chapter 14 is introductory. In fact, the author is convinced that the topics of this chapter should be investigated in more details to obtain some really deep results. For more references related to Chapter 14, we mention here Bank [28] and Rubel [1]. Essentially all results given in this book can be found in the original literature, at least in a closely related form. However, a number of proofs have been modified to make them more easily accessible to a non-expert reader. Straightforward computations have been mostly omitted. In fact, in easy situations, this should bring no difficulties to the reader, while modern symbolic software may always be used to check the more complicated cases. Concerning our extensive reference list, we have given a minimum number of references only from function theory and other areas of mathematics. In fact, we give nothing more than those references which have been strictly needed in our argumentation. On the other hand, we have tried to include as many articles from the global theory complex differential equations and closely related areas as possible. For the convenience of the reader, we have included the necessary review information to almost all items, mostly based on Mathematical Reviews and, occasionally, on Jahrbuch der Mathematik or on Zentralblatt für Mathematik. Since we have tried to compile an exposition which gives a reasonably easy access to the value distribution theory of solutions of complex differential equations, some aspects of recent research have been omitted intentionally. Such areas are, for instance, the following ones: (1) Most of the recent results based on Phragmen-Lindelöf type advanced arguments, see e.g. Bank and Langley [2]-[4]. (2) Factorization theory of meromorphic solutions of complex differential equations. In this area a good reference book recently appeared, see Chuang and Yang [1], The reader should observe that factorization theory in itself presents an excellent example of the applications of the Nevanlinna theory. (3) Algebroid solutions of complex differential equations, see He and Xiao [5] which creates, however, language problems to many of the eventual readers. (4) Tsuji value distribution theory applications into complex differential equations, see e.g. Rossi [3], (5) The recent research related to the asymptotic integration of complex differential equations, due to several mathematicians by the end of eighties, including the important papers by Brüggemann [1], [2] and Steinmetz [20], (6) The applications of the Strödt theory into the value distribution of solutions of complex differential equations, see e.g.

4

Introduction

several papers by Bank. Many of these omitted areas are still, at least partially, under strong development. One should perhaps wait some years before it can be seen how an exposition about these topics should be compiled.

Chapter 1

Results from function theory

This chapter contains some background material which seems to be of frequent use in the theory of complex differential equations. However, the Nevanlinna theory and the Wiman-Valiron theory will be considered separately in the next two chapters.

1.1 Two lemmas from real analysis These two elementary lemmas find their use when exceptional sets typical in the Nevanlinna theory have to be avoided from the conclusions. Lemma 1.1.1. Let g : (0, + o c ) —• R, h: (0, +oo) —> R be monotone increasing functions such that g(r) < h(r) outside of an exceptional set Ε of finite linear measure. Then, for any a > 1, there exists rg > 0 such that g(r) < h(ar) for all r > r0. Proof Denote σ := fEdr, and choose rο = σ/(α interval [r,cer] meets the complement of E. In fact,

— 1). For any r > rq, the

Therefore, taking t G [r,ar} \ E, we get g(r) 0 such that g(r) < h(ra) holds >

Proof Denote now Λ := JEdr/r < oo, and choose rg = exp(A/(a — 1)). For any r > rg, the interval [ r . r n ] meets the complement of E. As in the preceding

6

1. Results from function theory

proof, — = l o g r Q - logr = ( a - l ) l o g r > ( a - l ) l o g r 0 = Λ. Therefore, taking now t Ε [r, r a ] \ E, we get g(r) A — 1. Then the canonical product

fU(-) defines an entire function having zeros exactly at the points an, with prescribed multiplicities. • The above two theorems may now be combined to obtain the Weierstrass factorization theorem: Theorem 1.2.4. Let f be an entire function, with a zero of multiplicity m > 0 at Ζ = 0. Let the other zeros o f f be at a\, «2. · · • » each zero being repeated as many times as its multiplicity implies. Then f has the representation

f ( z ) = e ^ z

m

f [ E

m

n

( ( 1 . 2 . 2 )

η.

for some entire function g and some integers mn. If (an )n(=p;j has a finite exponent of convergence A, then mn may be taken as k = [λ] > A — 1 in (1.2.2). • Recall now that a(f)

:=

limSup'0gl0|8M(r-/). r—»-3G log r

(1.2.3)

where M(r,f) :— max| 2 | = r l/"(z)|, defines the order of an entire function. We add two remarks concerning the above theorem: (1) If an entire function / has a finite exponent of convergence A (f) for its zero-sequence (a n ) n( zf$, and we write the representation (1.2.2) in the form f{z)

=

Q(z)e*k\

applying Theorem 1.2.4, then we have A(Q) = &(Q) = A(/"), see Ash [1], Theorem 4.3.6.

8

1. Results from function theory

(2) If the entire function/ in Theorem 1.2.4 is of finite order σ, then g in (1.2.2) is a polynomial of degree < σ. This is the essential contents of the Hadamard factorization theorem. A couple of other elementary results concerning the exponent of convergence are frequently used. To this end, consider the sequence ( a n ) n e ^ from Definition 1.2.2. Denoting now n(t) := card((a„)„ e N Π {\z\ < i}), Γ N(r):=

(1.2.4)

n{t)dt - 4 t— ,

Jo

(1.2.5)

we recall, see Hayman [2], Lemma 1.4, that the series in (1-2.1) and a+l the integral J^°(n(t)/t ) dt converge simultaneously. Therefore, the exponent a so of convergence of (an)„ 0 JO

^

L

j

\

a

n \ ~

a

n(t) dt I ^ r r < ° 4

d·2·6)

Moreover, it is easy to prove by the above notions, see Boas [1], Theorem 2.5.8, that A = l i m s u p 1 ^r . r—>oc log

(1.2.7)

An immediate consequence of (1.2.6) is now the following most useful Lemma 1.2.5. Let (an)ne^ be a non-empty sequence of non-zero complex numbers (not necessarily distinct) with a finite exponent of convergence A. Then for every ε > 0, N(r) = Proof

0(rX+£).

Given ε > 0 and r > 0, we see by (1.2.6) that Γ n(t)dt < f°° n(t)dt / -χ+ε+] Ττ—τ -ΎΓ1Ύ Jo ί - Jo ίΛ+ε+1

:

=

κ

. . vε ; < °o.

Therefore, the integrated counting function (1.2.5) satisfies \

Γ n(t)dt

Γ

\ ,, n(t) dt

fr n(t)dt

. ,

λ,Ρ

L e t / be an entire function, or, more generally, meromorphic, and let ( a n ) n e j u j denote its sequence of zeros, each repeated according to its multiplicity. Denote

1.3 Complex polynomials

9

by A ( f ) the exponent of convergence of this sequence of complex numbers. Then it is easy to obtain the following Lemma 1.2.6. Let f\, fn be two entire functions with no common zeros. Then for Ε — f \ f z we get A(E)=max(A(A) ; A(/- 2 )). Proof. The inequality max(A(/j), λ(/2)) < λ ( E ) is trivial. To prove the converse inequality, we may assume that A ( E ) — A > 0. Then, for any ε > 0, the integral f

x

n(t) dt

Jo ^

^

diverges, with n(t) being the unintegrated counting function (1.2.4) for the zerosequence of E. With the corresponding notations n\(t), η2{ί) f ° r / i a n d / 2 , we have n(t)

=nl(t)+n2(t).

Therefore, at least one of the integrals [χη2(ί)άί

f°° ni(t)dt Jo

Jo

/λ+1~ε

diverges which means that m a x ^ ) ^ ) )

> A — ε.

Since ε is arbitrary, we have the assertion.

•

1.3 Complex polynomials For the convenience of the reader, we recall here two useful results concerning complex polynomials. The first one is the following self-evident lemma, see Jank and Volkmann [3], Satz 1.2: Lemma 1.3.1. Let Ρ {ζ) = anzn +αη_]ζ"~ι +• · ·+α$ with an φ 0 be a polynomial. Then, for every ε > 0, there exists ro > 0 such that for all r = |z| > tq the inequalities (\-e)\an\r" hold.

\an\\z\n(\ j== ι1 3

V

= M\z\n

>

i -

Μ kT^i

\αη\\ζ\η(\-ΜΣ\ζ\~ή

J

=

3= 1

V

\an\\Z

J

>n(\z\-\-M

'I

lzl-1

Hence, if |z| > 1 + M, we have |P(z)| > 0 . Therefore all zeros of P(z) in |z| > 1 must satisfy the inequality (1.3.1). On the other hand, all zeros of P(z) in |z| < 1 satisfy (1.3.1) trivially. •

1.4 The Wronskian determinant This section contains a number of basic properties of the Wronskian determinants. However, we don't list properties which are just elementary facts of general determinants. We shall follow rather closely the presentation given in the thesis of G. Hennekemper [1], p. 11-19. Definition 1.4.1. The Wronskian determinant W (f\,... functions /], . . . , fn is given by fx •··

W(fu...Jn):= («-!) 1

• · ·

,/„) of the meromorphic

fn f Jn An-1) Jn

1.4 The Wronskian determinant

11

Moreover, we denote, for u = 0, . . . , η — 1, by Wv(fU...Jn) the determinant

which

comes

from

W (f\,...

,fn)

by replacing

the row

( f i " \ . . . , f i v ) ) by ( f i n \ . . . j i " ] ) . Proposition 1.4.2. Let J), ... , fn be meromorphic functions. Then W (/i,... ,/„) vanishes identically if and only if f\, ... , fn are linearly dependent (over C). Proof. The sufficiency being trivial, let us assume that W(f\,... ,fn) = 0 . We may also assume that f\ does not vanish identically, hence W(f\) ^ 0. Therefore, there exists ν £ {I,... ,n — 1} such that ,... ,fv) φ 0,

W(f],...Ju+[)

= 0.

(1.4.1)

Let now a e C be chosen so that f\, ... , fu+\ don't have a pole at a and that W(fi,... Jv){a) φ 0. By (1.4.1), the column vectors of W (/"],... Jv+\) are linearly dependent. Hence, for some constants C\, ... , Cv G C,

Hence, defining (1.4.2) we get ui-fl\a)

= 0

for μ = 0, . . . , v.

Moreover, by (1.4.1) and (1.4.2), (1.4.3) Expanding W(f\,...

, f v , u) by the last column, we obtain by (1.4.3)

(1.4.4)

12

1. Results from function theory

By repeated differentiation of (1.4.4), we see that u ^ ( a ) = 0

for μ £

NQ.

By the elementary uniqueness theorem of meromorphic functions, u (1.4.2),/j, . . . ,/ί,+ i are linearly dependent.

Proposition 1.4.3.

Let

complex

Then

numbers.

f\,

...

, f

(a)

W ( c ] f \ , .. . , C n f „ ) =

(b)

W

(c)

w ( f

(d)

W ( g f

(e)

W ( f

..., u

l

. . . j l

n

( η -

, g be meromorphic

functions

C1 · - - C n W i f x , . . . J

,g

1)!

=

8

n

and

c\,

• . . . ,

cn

be

) .

»

, i ) = ( - i y w ( / ; , . . . , ύ ) .

, . . . , g f

, . . . , f

n

0. By

H

n

) = g

) = f

i

n

W ( f

n

l

, . . . J

W (

n

) .

< · · · > ( ! )

Proof,

(a) This is nothing else but a basic calculation rule of determinants. (b) This follows immediately by repeated determinant expansion according to the first column vector. (c) Expansion according to the last column vector. (d) A standard reasoning shows that gfn

gf\

W ( g f

[

, . . . , g f n ) =

gf\

···

gfn

(1.4.5)

holds for u = 0, . . . , η - 1. In fact, for ν = 0 this is just the definition of W ( g f \ , . . . , g f ) . To prove the inductive step, we may assume that f \ , . . . , f , g , g', ... , / 0, oo at a given point a e C. Since n

n

.7=0

^

J

'

holds for i = 1, the induction at a Ε C follows by subtracting the first u + 1 rows, multiplied by (^j" 1 ) { g ^ ' ^ g ' 1 respectively, from the (is+2) t h row

1.4 The Wronskian determinant

13

of (1.4.5). The final inductive conclusion now follows by the elementary uniqueness theorem of meromorphic functions. For ν = η — 1 we therefore obtain gfi W{gfi,...,gfn)

gfn

gnW(fh...jn).

=

=

(e) By the preceding parts,

• Proposition 1.4.4. Letf\,

... , fn be meromorphic functions. Then

-rW(f[,...Jn) dz

=

Wn_[(fu...Jn).

Proof. Applying the standard Leibniz formula, see Kowalsky [1], p. 87, to define a determinant we obtain d dz

7

VeS,, £

Σ

+

Σ

(

σ

Κ ( \ / σ ( 2 )

ε

(

σ

-

' •/j(n),) +

£

Σ

)/σ(ΐ/σ(2) ' •·4'(η-\/σ(η)

/; · /; ·

/," ·

· ·

••

fk

/l

&

f"

ft

· •

•

An-1) Jn

+

·· ·

/," · f(n-1)

•M

] )

+

Σ

ε

^ σ { \ / σ { 2 )

Λ

f\

f" Jn f" Jn

f[

An-1) ••

' ""/j"«)0 +

^ M \ ) f ä ( 2 / a ( 3 )

Jn

' ' '

' ' 'f a { n - \ j a { l )

.·

fn

•

+ ••• + An•M

•M

2) •

·

An-2) Jn A") Jn

= w,>_,(/,,. ..,/„).

•

Proposition 1.4.5. Let f\, ... , /„ be meromorphic functions. Then, for η = 2, we have W(fuf2)

=

W(fi)w'(f2) - W(f2)w'(f,) =/1/2/-Ml

14

1. Results from function theory

and for η > 3,

W(fli...Jn-2)W(fl,...Jn)

= W(fl,...J

n

_l)-W(f

U

...J

n

^2J

n

)

Proof. The case of η = 2 is trivial. For η > 3, we see that

/l

···

fn-2

···

An—3) Jn-2

fn

(0) An—3) JI

/l j

(0)

,(n- 2)

An-2) Jn-2

J1 f(n-

Jn-2

fn-2

f\ JnJ\

An-1) J\

fl

An-3)

-3)

Jn-2

··

·•

fn— 1

··

4-1

··

An-3) J1

An-

/l

0

Jn-1

,(n-2) 4-1 f(«-i) 4-1

An-2)

An-1)

1)

An-3) ·•

J1

0

/n-1

An-3) J\

(«-3) /n

,(«-2) Jη

An-1) Jn

fn

3) An-3) Jn

0

fn — \

(1.4.6) (0)

AnJ1 An-

J1

-2) -i)

...

,(«-3)

An-3)

·'

•'«-I

•·

Jn-1

(n-2) 4-2 J1 („_,) ( n _l) 4 - 2 -M

•·

0 («-2) Jn

(κ-1) An-1) 4 - 1 Jn

by an apparent row addition operation. We now apply the Laplace expansion rule (Kowalsky, [1], p. 92) to the equation (1.4.6), for the (n - l) t h to (2η - 4 ) t h rows on the right hand side and the (n — l) t h to (2n - 3) th columns on the left hand side. This results in

1.4 The Wronskian determinant /l

fn-2

An-3) J1

f

Jn-2) J n—\

•••

An-2) 7l

An-1) · · · Jη

·••

(n-3)

(,i-2)

f\

An-\) J\

f\

fn — 1

fl

fn

f\

An-3) Jn-2

···

fn-2

fn

•··

,(«-3) r{n—2>) Jn-2 J'1 ,(«-l) Jn Λί-2

fl

fn — ]

•••

(η-1) [

fn-2

fn An-3) • Λί-ι «-η J n

(«- 3)

r{n—2) An-2) · · · /n_2 Jn

(n-1)

•

which is just the assertion. Proposition 1.4.6. Let f\, and define

15

fn be linearly independent meromorphic l

anM:=-{W(fh...,fn))

W„(fu.

..,/„)

functions

(1.4.7)

for ν = 0, . . . , η — 1. Then the meromorphic functions any have their poles among the zeros ofW(f\,....fn)or the poles o f f \ , ... , fn. Moreover, l

a„.n-\=-(W(fu...jn))

-^W(fu...Jn).

Finally, f\, ... , fn satisfy the homogeneous linear differential

(1.4.8)

equation

η—1 (1.4.9) 0 Proof The first assertion is straightforward. Next, (1.4.8) follows from (1.4.7) by Proposition 1.4.4. To prove (1.4.9), let h be any meromorphic function. Expanding W{f\,... ifn-h.) according to the last column we get W(fu...Jn,h) W(fu...Jn) The assertion (1.4.9) now follows immediately.

η-1 Μ

(1.4.10)

ι/=0

•

16

1. Results from function theory

Proposition 1.4.7. Let f\, ... , fn be linearly independent meromorphic satisfying a homogeneous linear differential equation

functions

n-l +

(1.4.11) v=o

Then av = - ( W ( f u . . . , / „ ) ) " '

, . . . ,/„).

(1.4.12)

Proof Denoting anM :— 1 and an :— 1, we may consider (1.4.9) and (1.4.11) as linear system of equations to determine an^(z), resp. a^(z), at a given z. Since W (f\,... , f n ) φ 0, such a linear system has a unique solution, and therefore we get (1.4.12) for ν — 0, . . . , η — 1, at least outside of zeros of W(f\,... ,f„) and poles of the functions in consideration. By the elementary uniqueness theorem of meromorphic functions, (1.4.12) holds in the whole complex plane. • Proposition 1.4.8. Let f\, ... , fn be linearly independent meromorphic of f

{ n )

+ a

n

- M f ^ + ---+a0(z)f

solutions

= 0

with meromorphic coefficients. Then the Wronskian determinant W if\,... ,fn) satisfies the differential equation W' + an_\{z)W = 0. Specially, if an_\ is an entire function, then for some C £ C, W (f\,..., fn) = C exp φ where ψ is a primitive function of —an _ ι.

Proof. Since = - a n - i W f - *

aMf

for i = 1, . . . , n, we see by Proposition 1.4.4 that fi f\ W' = —W(flt...Jn) dz

= Wa_l(fl,...

Jn)

=

Jn-2) J] Jn) J\

••• ·• •

fn fn

Jn-2) · · · Jn Jn) Jη

·

·

·

1.4 The Wronskian determinant •

f n

f' Jn

// («'- 2)

An-2)

• J/2 («-1)

- then

(

n

s

, η

Παι·) =ΐο§( Γ Μ

/=1

'

i=\

(f) By (b) and (e) above,

η

(

η

\

'

η

=Σ1ο§α'·

i= l

i=1

χ

η

a,· j < log^« max a
2N(r, \

1

λ

f - a

hence & ( a j ) = 1 - limsup r-oc From (2.5.2) we immediately conclude

) r /_J - a J ^> 1 T(r,f)

2.6 The Ahlfors-Shimizu characteristic function

49

Corollary 2.5.6. A non-constant meromorphic function f admits at most four completely ramified values. •

2.6 The Ahlfors-Shimizu characteristic function In some order considerations, the Ahlfors-Shimizu characteristic function, measuring the weight of the image of / on the Riemann sphere, is more suitable than the usual Nevanlinna characteristic function. Omitting the proofs, which may be found in Hayman [2], p. 10-13, we just mention the basic definition and the correspondence between these two characteristic functions. In fact, the Ahlfors-Shimizu characteristic function will be defined by

ro(r,/):= JoriMu, t

where Wo

Jo

(1 +

\f{Qe^)\r

Now, an essentially geometric analysis on the Riemann sphere results in U r j ) = ^

ζ *

log φ

+

- log yj\

+ \f(0)\2

+N(r,f).

From this it is rather obvious to see that the difference of 7o(r,/) and remains bounded (by a constant independent of r), provided /(Ο) Φ oc.

T{rJ)

Chapter 3

Wiman-Valiron theory

The Wiman-Valiron theory is an indispensable device while considering the value distribution theory of entire solutions of complex differential equations. Fortunately, the monograph by G. Jank and L. Volkmann contains an excellent presentation, see Jank and Volkmann [3], p. 30-38, 187-199. Therefore, we restrict ourselves to give here just a short review of basic notions and most important results. The difference of what follows below to [3] is that we consider entire functions g : C —> C only, while [3] considers vector-valued functions g : C —> Cn whose components g], . . . , gn are entire functions. Let now g be an entire function whose Taylor expansion is oc

g{z)

Yta„zn.

= n= 0

Clearly, the power series Y^=o\an\rn given r > 0,

converges for every r > 0. Then, for a

lim \an\rn n—> oo

= 0,

and the maximum term μ(Γ) = ji(r,g)

: = max n> 0

\an\rn

is well-defined. This makes it possible to define the central index v{r) = v{r,g) as the greatest exponent m such that \am\rm = ß(ryg). Clearly, for a polynomial P ( z ) = a n z n Η l· a 0 , a n ^ 0, we have ß(r,P)

= \an\r",

v(r,P)=n

(3.1)

for all r sufficiently large. In the general case, \an\rn

< ß(r,g)

for all η > 0,

n

< ß(r,g)

for all η > v{r,

\a„\r

g).

3. Wiman-Valiron theory

51

Because of (3.1), we may assume that g is a transcendental entire function while considering basic properties of the maximum term and the central index. Here it is enough to recall that (1) ß(r,g) is strictly increasing for all r sufficiently large, is continuous and tends to +oo as r —> oo; (2) v{r,g) is increasing, piecewise constant, right-continuous and also tends to +oo as r —• oo, see Jank and Volkmann [3], p. 33-35. In applications to complex differential equations, two results are of major importance: Theorem 3.1. If g is an entire function of order σ, then log u(r ,g) loglog^(r ,g) σ = lim sup —— = lim sup -. r —>oc log r r—»oc log r Proof. See Jank and Volkmann [3], p. 36-37.

•

Theorem 3.2. Let g be a transcendental entire function, let 0 < δ < ^ and ζ be such that I ζ | = r and that \g{z)\>M{r,g)u{r,g)-*+S holds. Then there exists a set F C M+ of finite logarithmic measure, i.e., fF dt/t < +oo, such that

holds for all m > 0 and all r £ F. Proof. See Jank and Volkmann [3], p. 187-199. The above entire solutions techniques can one can find in

•

two theorems form a powerful tool for order considerations of of linear (and algebraic) differential equations. The corresponding be found below in their natural connections. In addition to what Jank and Volkmann [3], we shall need the following proposition:

Proposition 3.3. Let g be a transcendental entire function of order a(g) = 0. Then, for all k e N, / € N, lim

= 0.

52

3. Wiman-Valiron theory

Proof. Clearly, it suffices to prove that

lim r—*oo

=

r

0.

(3.2)

Take ε > 0 small enough so that ke < 1. By Theorem 3.1, we obtain a(g)

=

lim r _ > o c (logz/(r, g ) / l o g r ) = 0, and so log v(r,g)k

= k logi/(r,g) < k logv(r,g)

< kelogr

=

logrke

for all r sufficiently large. Therefore v{r,g)k r Since ke — 1 < 0, we obtain (3.2).

r^_ ~ r


· · · > an-i{z)More precisely, the results in this section have been selected to motivate some more specific questions connected with linear differential equations, to be considered in the subsequent sections. As is well-known, all solutions of (4.1) are entire functions, see, e.g., Herold [1], Satz 1.3.2. Perhaps we should comment this usual but somewhat inexact assertion that all solutions of a differential equation are entire functions. This is, of course, nothing but an abbreviation of the more precise statement that all local solutions admit an analytic continuation into the whole complex plane, meaning by the monodromy theorem that an entire global continuation exists. Concerning the classical local existence theorems, see Herold [1], Kapitel I. The fact that the global continuations of local solutions still satisfy the same differential equation is an immediate consequence of the elementary uniqueness theorem of analytic functions. We begin with the following "classical" theorem due to H. Wittich, see Wittich [15], Satz 1. Observe that the application of the Wiman-Valiron theory, appearing in the proof below and in some later proofs, will be carried through in all details, in order to familiarize the reader with this method. Later on, we gradually proceed to apply the Wiman-Valiron method in its usual short-cut form. Theorem 4.1. The coefficients ÜQ, ... , an_\ of (4.1) are polynomials if and only if all solutions of (4.1) are entire functions of finite order. Proof. Assume first that the coefficients üq, . . . , an_\ are polynomials. L e t f ( z ) be a transcendental solution of (4.1) and let v{r) be its central index. By Theorem 3.2, let F C M + be a set of finite logarithmic measure such that

(4.2) holds for ι = 0, . . . , η and for r = \z | ^ F, ζ being chosen as in Theorem 3.2. For instance, we may assume that \f(z)\ — M(r,f). Substituting (4.2) into (4.1)

54

4. Linear differential equations: basic results

we obtain

+ α ι ( ζ ) ^ ( ΐ + o ( l ) ) + e o ( z ) ( l + o ( l ) ) = Ο,

since (l + o ( l ) ) / ( l + o ( l ) ) as well as 1 / ( 1 + o ( l ) ) are both of type 1 + o ( l ) . Hence we get u(r)

n

+za

n

_l(z){l+o(l))u(rr-l+.·. + z

n

-]al(z)(l+o(l))u(r)+z

n

a

0

( z ) { l + o ( \ ) ) = 0 . (4.3)

Denoting Π) Qn-i{z)

l

:= z a„-i(z)

Σ€ν*'>

= 3= 0

we have to consider v{r)n

+ Qn_!(z)(l

+ o{\))v{r)n~x

+ · · · + Q0(z){

1 + o{ 1)) = 0 .

If r φ F is sufficiently large, we may assume, by Lemma 1.3.1, that for these values of r we have \Qn^(z)(l+o(l))\ni\rn',

i = 1, ...

, n.

By Lemma 1.3.2, we must have u{r) < 1 + max | ö n - , - ( z ) ( l + o ( l ) ) | < 1 + 2 max (|c I > ( \r n >). 1 1 is arbitrary, we see that

To prove the converse assertion, let the coefficients üq, . . . , an_\ of (4.1) be entire functions and assume that (4.1) possesses a solution b a s e / i , . . . , / « of entire functions of finite order of growth. Clearly, the Wronskian determinant

W =

W(fl,...Jn)

fn f' Jn

Δ f( An-1) J\

7^0

An-1) · · · Jn

is an entire function. By elementary order considerations, W is of finite order of growth, see Definition 2.1.12 and Remark to Corollary 2.3.5. We may now express the coefficients ag, . . . , an-\ in terms o f f \ , . . . , f n . In fact, by Proposition 1.4.7, an-q(z) = -Wn-q(fi,... ,fn)(z)/W(z), q = 1, ... , n, see also Definition 1.4.1. By elementary order considerations again, all coefficients üq, . . . , α„_ι must be of finite order. We now apply the standard order reduction procedure by substituting

into (4.1). Let Vj ^ · = / / / ι , i.e., (v|

= vj, and define an := 1. Then

k = 0, ... , n.

(4.4)

56

4. Linear differential equations: basic results

Substituting (4.4) into (4.1) we obtain

k m) °=Σ> Σ ( )f^r' =j = -Σ "£' m k= 0 m= 0 1 m=0 V

= vf-

7

+

V

7

1

^ ^ + « » - ΐ / ι ί Λ - 1 ) + ·•• + «!// + α ο Λ ) + / ΐ ν 1 ( η - Ι ) n-2/i-j-l / · . , , s

+Σ Σ f+ j=0 ™=o

^wi·».?1·

v

7

Since/i solves (4.1), the v | ^-term vanishes. Dividing b y / ] we obtain v i " - 0 + ahn_2(z)v[n-2)

+ •••+

alfi{z)vi

= 0

(4.5)

where n-j-1

a

1 J

=a

J + i

+

/ · , , .

\

^

Jöj+1+m-^—

m

Am)

(4.6)

for j = 0, . . . , rt — 2. By elementary order considerations again, a\ o, ... , a\,n-2 are meromorphic functions of finite order. Moreover, the meromorphic functions

are solutions of (4.5) of finite order of growth. To prove that v^i, . . . ,v\ n _ \ form a solution base to (4.5), it remains to show that they are linearly independent. But if for some C\, ... , Cn _ ] G C we have C\vM

+

l·

Cn_ivi)n_i

=0,

then d f C]fj Η dz\

/,

\-

Cn_\fn

Hence there is a constant Co G C such that -Ctfx

+ C,/ 2 + · · · + C„_]/„ = 0.

Since f\, ... , fn are linearly independent, we must have CQ = C\ = · · · = C„_i = 0.

4. Linear differential equations: basic results

57

Next we show that m { r , a

l

:

j

)

=

0 ( l o g r ) ,

j

=

0,

. . .

, η

-

2,

(4.8)

implies = O(logr),

m ( r . a . j )

ι = 0

« — 1.

(4.9)

In fact, a

l . n - 2

a

=

n - 1

n

+

>

J1

implies

m ( r , a

n

- \ )


0 and 5 G C, hence zero is clearly a Picard value for all members of the corresponding solution base f\, . . . , f n of (4.1). Assume now that a solution base / i , . . . ,/„ of (4.1) is such that every f j has zero as its Picard value, i.e., fj(z)=Pj{z)e^zl

j=l,...,n,

(4.23)

holds for some polynomials pj, qj. The case η = 1 is immediate. In fact, from f ' ( z ) / f ( z ) = — ao(z) we see that / has no zeros. Hence, any polynomial q such that q = q\ + ηζ, 7 £ C \ {0}, gives the desired transformation. To prove the assertion for η > 2, we consider the Wronskians f o r / ] , . . . ,fk-\,fs for all .v and k such that 1 < k < s < n. By (4.23) we obtain W(fu---,fk-l,fs)=Pueqi+""Hlk-l+il'

(4-24)

and W'(fu.

· · J Ä - b / v ) = (PkAq[

+ · • • + q'k-\ +qs)+P'k,yi+'"+qk-l+Cls

(4-25)

where P^ s is a polynomial. By Proposition 1.4.5, we have W(fi,....fk-i)W{fl,...,fkJs) =

W(fi....Jk)W'(fii...Jk_ufs)-W(fl,...,fk_lJs)W'{fi,...,fk).

66

4. Linear differential equations: basic results

Substituting here the corresponding expressions from (4.24) and (4.25) we obtain Pk.hk.lPk+h5

= Pk,kPk,s{qs

~q'k) + Pk.kP'k,s ~ pikpk,s

(4.26)

for A: > 2 , 5 = k + 1, . . . , η. For k = 1, s = 2, . . . , η, we obtain similarly Pl,s =P\Ps{q's - q[) + PiPs ~ p\ps-

(4.27)

We denote now: «ΐ,ί : = deg {pi), ßü •

=

i = 1, ...

,n\

j — I, ... , η —

deg(q- — q'j),

j = 2, ... , n;

= deg (Pj,i),

i = j + 1 , . . . , n;

(4.28)

i = j, ... , n.

After this first part of the proof, we proceed to establish the following Lemma 4.6. Suppose the sequence q'^...,q'n of derivatives of qi from (4.23) contains r distinct functions, r < n, such that each of the distinct functions appears m\ times, while m\ +••·+ mr = η > 2. Then η

η—1 η

i'=l

,i + Σ Σ ß.« j=1 i=j+1

®n,n =

r - Ϊ)·

(4 29)

"

i=l

Proof. The idea below is to express the degrees aks in terms of a\ j and ßjj. The assertion then follows by taking k = s = n. Since we consider the degrees of polynomials only, we may assume, if needed, that \z | is large enough. The proof will now be divided in three sections. (1) Assume first that m\ = · · · = mn — 1, i.e., that for all i, j = 1, . . . , n, i φ j, we have q'· ψ q'r Denoting α^ο = (4.26), (4.27) and (4.28) immediately result in the recursion formula a

k+\,s =

+ ak>s + ßk>s -

(4.30)

for £ > 1, s = £ + 1, . . . , η. We now assert that k-1

k-2 k-1

1 , 5 — k, ... , n. Observe here and below that a sum of type Σ)·' = , 0 considered to disappear whenever i\ < For k — s — n, (4.31) reduces back into (4.29):

η —1 an,η

a

= Σ

n—2 +

U

ι=1

η

Σ

—1

Σ

j=\

η—1

ßj-i + " Μ

+

i=j+\

Σ 1=1

η =

Σ

+ • • * + A , « - l ) + (/?2,3 + * · * +

"

/=1 η

=

Σ

a

h l

+ { ß

h 2

+

* · · + ßl,n)

+ {ßl,3

+

•··+/%,») +

• · • + ßn — \.n

i—\ η

=

n—\

η

+ Σ ι= 1

Σ i=j+l

Now, for k — 1,5 = 1, . . . , η, (4.31) holds trivially. For k = 2, s = 2, ... , n, we obtain

a

CÜ2,s = "1,1 +

\,J +

= «1,1 +

a

l , s + A,5 -

a

0,0

from the recursion formula (4.30). But this is nothing else than (4.31) in this special case. Assume now that we have proved

r —1

ar,j = Σ

r—2 a

u + Σ

r—1

Σ

r—1

ßjj +

+ Σ

for*

(4·32)

for 1 < r < k — 1, s = r, . . . , n. By the recursion formula (4.30) we obtain, using (4.32),

68

4. Linear differential equations: basic results

α*.λ = ak-\.k-\ k-2

+ α*-1,.,· + Ä t - U -

ak_2±-2

it-3 it-2

α = iΣ Μ +j=lΣi=j'+l Σ ^ =1 lt-2

it-3

+ a-l.s + Σ

+

/=1

it-1

+

A—3 )t-2

)t-2

it-2

i=l

+i' = l

a

it-4 it-3

-

~Σ M Σ Σ ^ /=1

j=l/=j+l

It-3 it-2

+j=Σ1 /=j+1 Σ fa _

it-3

~Σ

;=1

it-2

α

= 1Σ= 1 ^ +j=Σ1 ,·=/+1 Σ ^ + (=1 it-3

it-3

k 2_

it-1

+ Σ ßj- ~ Σ + - ν + Σ ^ j=1

it-1

α

i=l

it-2 it-1

= Σ ι- + Σ Σ i=l

j=lI=j+1

(=1

it-1

+ Σ^'-ν 1=1

which is just (4.31). (2) Suppose now that at least one of m\, ... , mr is > 1. We may assume that the solution base (4.23) has been arranged in such a way that all possible repetitions of a given q j come immediately after the first occurrence in the sequence, while all derivatives qj appearing only once are before repeated derivatives qj. In other words, derivatives q- will be grouped in r blocks, each having equal members qand different blocks having different derivatives q[, while blocks of length = 1 begin the whole sequence. Moreover, we may assume that the corresponding numbers «1 ,· = deg(pi) form a strictly increasing sequence inside of each block. This may be achieved, if necessary, by replacing the functions f j by suitable linear combinations of them, which obviously does not destroy the assumed Picard property. In fact, if there are several equal degrees α ι , , we may reduce these degrees by linear combinations for all but one of these solutions. Continuing down to the lowest degree we get the desired situation. Let now q'm, ... , q'm+t be the first block of length > 1. Since q[, ... , q'm are all distinct, Part (1) remains valid and we see that (4.31) still holds for 1 < k < m, s = k, ... , n. Observe that this is also true for m = 1, i.e., in the case that all blocks are of length > 1. Also in this case, we apply ao.o = 0, if needed. Proceeding now to compute a^ s for m + 1 < k < m + t + 1, s = k, ... , n, the use of (4.26), hence of the recursion formula (4.30), changes since q's -q'k =0 for m < s,k < m + t. In this case, the degree of Pk.k^'k s- - ^jt has to be

4. Linear differential equations: basic results

69

computed. To do this, we first observe that m—1 ft«..? -

^

m—2 m— 1

qι,/ + y ^ s~m>

~~

a[ m

-

1

for 5 = m + 1, . . . , m + t, since a\ ,· increases strictly along with i and ß, s = deg{q's -