Univalent Functions and Orthonormal Systems

Citation preview

Translations

of Mathematical Monographs Volume 49

Univalent Functions and Orthonormal Systems by

I. M. Milin

American Mathematical Society Providence, Rhode Island 1977

OLl,HOJU1CTHhlE «I.>YHKU1111 11 0 PTO HOPM11POBAHHhlE C11CTEMhl 11. M. M11Jl11H l13AaTeJlbCTB0

11

11ayKa"

MocKBa 1971

Translated from the Russian by Israel Program for Scientific Translations Translation edited by P. L. Duren AMS (MOS) subject classifications (1970). Primary 30A34; Secondary 30A31.

Abstract. This book is devoted to a method of investigating univalent functions by means of orthonormaf systems. Using this method, many of the recent results of Soviet and other mathematicians on the coefficient problem are presented. For univalent functions in finitely connected domains, conditions for univalence are found, as well as the range of various functionals.

Library of Congress Cataloging in Publication Data

Milin, Isaak Moiseevich. Univalent functions and orthonormal systems. (Translations of mathematical mg.pographs; v. 49) Translation of Odnolistnye fu.nktsii i ortonormirovannye sistemy. Bibliography: P• 1. Univalent functions. I. Title. II. Series. QA331.M5513 515 1 .93 77-1198 ISBN O-S2J.8-1599-7

Copyright © 1977 by the American Mathematic.al-Society

TABLE OF CONTENTS Preface......................................................................................................................... 1 Introduction.................................................................................................................. 2 Part I. Univalent functions in a simply connected domain.................................... 7 Chapter 1. The system of functions {An(z)}............................................................ 7 § 1. Connection with Faber polynomials............................................................ 7 §2. Area theorems................................................................................................... 9 §3. Orthonormality of the system {ynA~(z)}.................................................. 16 §4. Simplest applications of the systems {An(z)}............................................. 22 Chapter 2. Taylor coefficients of a composite function....................................... 27 § 1. Inequalities for the coefficients of a composite function in the general case........................................................................................................................ 27 §2. Inequalities for the coefficients of a composite exponential function... 32 §3. Asymptotic equalities for the coefficients of composite exponential functions............................................................................................................... 44 Chapter 3. Coefficients of univalent functions......................................................... 51 § 1. Inequalities for the logarithmic coefficients................................................ 53 §2. Asymptotic behavior of the coefficients...................................................... 64 §3. Bounds on the coefficients........................................................................... 74 §4. The local coefficient problem........................................................................ 82 § 5. Growth of coefficients of bounded functions............................................. 92 Editor's Supplement to Part I. P. L. Duren............................................................... 101 Part II. Univalent functions in a finitely connected domain................................. 105 Chapter 4. The Laurent system of functions......................... ................................... 105 § 1. Existence and uniqueness of the Laurent system....................................... 105 § 2. Properties of the functions of the Laurent system.................................... 116 § 3. Expansion in series in terms of the Taylor system.................................... 132 §4. Expansion in Laurent-type series.................................................................. 139 Chapter 5. The system of functions {Cn(z)}............................................................ 153 §1. Simplest properties of the system {Cn(z)}................................................... 153 §2. Area theorems................................................................................................... 157 §3. Orthonormality of the system {Cn(z)}......................................................... 162 §4. Extremal systems {Cn(z)}.............................................................................. 168 Chapter 6. Applications ................................................................................... :•···· .. ···" 179 § 1. Estimates and ranges of some functionals.......... ......................................... 179 iii

TABLE OF CONTENTS

§2. Conditions for univalence ...................... :..............................................._.......

191

Bibliography.................................................................................................................. Supplementary Bibliography.......................................................................................

195

iv

200

PREFACE

Univalent functions, i.e. regular or meromorphic functions which assume different values at different points of a domain, are the simplest analytic functions from the geometrical standpoint. Their investigation involves questions concerning criteria for univalence of analytic functions and the influence of this property on other properties of the functions. In the present book orthonormal functions are applied to the study of univalent functions, according to the following scheme. In Part I, which contains three chapters, we consider univalent functions in a simply connected domain. Attention is centered here primarily on the behavior of the Taylor coefficients of univalent functions; in the first chapter we study the properties of a special system of functions for the coefficients, while in the second we find sharp bounds and asymptotic equalities for the coefficients of a composite exponential function. The heightened interest in this question is undoubtedly due to the coefficient problem. Part II is devoted to univalent functions in a finitely connected domain containing the point at infinity. In Chapter 4 we construct the so-called Laurent system of functions, which plays the same role for an arbitrary finitely connected domain as the system {zn} for a circular ring domain. The fifth chapter is analogous to the first, and the last is devoted to applications, principal attention being paid to determination of the ranges of various functionals. Since the exposition in the second part of the book is independent of the first, the reader may proceed to Chapter 4 immediately after the Introduction. The author is indebted to G. V. Kuz'mina, who read the manuscript attentively and offered a number of remarks, and also to A. Z. Grinspan and V. I. Milin, who were of great assistance in preparing the manuscript for the press.

1

INTRODUCTION

l. Finitely connected domains and classes of functions. Let B be a finitely connected domain in the z-plane containing the point at infinity, whose boundary r is the union of m mutually exterior closed analytic Jordan curves r 1}, and its interior by K 0 : K° = {z: lzl < 1}. The Green's function for B 0 is G{z, 00 ; B) = lnlzl, and therefore its level curves r P are concentric circles lzl = p (p ~ 1) about the origin. Correspondingly, the domains BP, the sets Br R, etc. are simplified . The following classes of functions will be considered: L 2 (B)-the class of regular functions with square-integrable modulus in B; z2 {B)-the subclass of functions in L 2 (B) having a single-valued integral in B; a(B)-the class of functions f(z) regular in B, [( 00) = 0, for which the area of the image of Bis finite, i.e. for which /'(z) E z2 (B); L(B)-the class of functions F(z) meromorphic and univalent in the domain B, with a pole at infinity and Laurent expansion

.

(6) in a neighborhood of the point at infinity; f(B)-the subclass of L(B) consisting of the functions F(z) which map B onto a domain without exterior points whose boundary has zero area; L and f-the classes L(B 0 ) and f(B 0 ), respectively; _ LM-the subclass of functions F(z) E L such that IF(z)I >Min the domain B 0 , where O ~ M ~ 1 ; S-the class of functions f{z) regular and univalent in the unit disk, normalized by (7) SM-the subclass of functions f(z) ES such that 1/(z)I a*(F) = 1Tr2 , we have 0.

Consequently the function Q'(w) is regular in the closed domain JP,€ and has no zeros there. This at once implies the inequality

12

I. THE SYSTEM OF FUNCTIONS {An(z)}

cr(lp)=

~~ jQ'(w}l 2 dcr~ ~~ IQ'(w)j 2 d:

d] > 0,

p> I.

Since F(z) E ~o has no zeros inn°, the function

(F ;2> r· = exp { 11, lop ;2>}

is regular in lzl

> 1 and therefore, for p > 1,

00

=p2>. ~ IDn(t-)j2p-2n, n=O

We now conclude from (1.27) that, for any p 00

I

> 1,

~ (2t--2n) Dn (t-)12 p-2n n=o

> 0,

z=pei'f',

(1.27)

I. THE SYSTEM OF FUNCTIONS {An(z)}

14

or

00

~ (n-i..) f Dn (i..)12 p-2n

< i..,

n=l

whence, by letting p ~ I, we deduce (1.26). As for whether equality holds in (1.26), the assertion is proved just as in Theorem 1.1, since by (1.25) and (1.22) the left-hand side of (1.27) represents the area of the domain di obtained from the complement dP by the transformation of polar coordinates R * = R.,._, 0 * = 0. In connection with inequalities (1.17) and (1.23), we mention without proof (for the proof, see Prawitz [1]) that under the assumptions of Theorems

1.1 and 1.2 we have the more general inequality

~ [-12n 52"' Q (j Q(F(pel 0, dp 0

from which we can deduce inequality (1.17) (set Q(R) = R 2 ) and Theorem 1.2 (set Q(w) = w). · It is worthy of attention that the integrand Q(IQ(w)I) is in all cases subharmonic in the domain dPo· Indeed, the function ln R = lnlQ(w)I is subharmonic in dPo (although it is discontinuous at the zeros of Q(w)), and Q(R) is by assumption an increasing convex function of ln R. Hence the composite function Q(IQ(w)I), being continuous, is also subharmonic in dPo (see, for example, Timan and Trofimov [ 1] , pp. 186-187). Nehari [I] recently showed that this fact is not accidental. The following theorem is a slight modification of Nehari's results, preserving the main idea of his proof.

1.3. If F(z) E ~ and S(w) is a continuous subharmonic function on the complement dPo' Po> 1, then the mean value THEOREM

s

21'

H(p) = 2~

S (F(pe1ff)) dcp

0

is a nondecreasing convex function of ln pin the interval 1 < p if the function H(p) is differentiable, then

H' (p)~ 0,

< p0 .

Hence,

(1.28)

with equality for P = p' E (1, p0 ) only for functions S(w) harmonic in dp'· We first show that any function U(w) harmonic in dP satisfies (1.28) with the equality sign. In fact, since dPo is a simply connected ~amain, it follows that U(w) can be expressed as the real part of some function T(w) regular in d Po U(w) = Re T(w). The composite function T(F(z)) is regular in the annulus

§2. AREA THEOREMS ·'

1 < lzl

15

< Po, and therefore, by the Cauchy integral theorem,

s

~ 2m

T (F (z)) dz = ~ z 2m

lz1=P1

s

1

T (F (z)) d z Z,

< P2 < P1 < Po·

Jzl=P,

It follows from the last two equalities that ll:rt

2~

SU (F (pe1t)) dq> = const,

1 < P < Po,

(1.28a)

II

as required. Now let S(w) be the given subharmonic function; we shall prove that H(p) is monotone nondecreasing, i.e. (1.28b) for arbitrary p 2 < p 1 in the interval (1, p 0 ). To do this, we define a function U(w), harmonic in dP and continuous in 1 dp 1 , with boundary values on CP 1 equal to S(w). That this is possible follows from the existence and uniqueness of the solution to the Dirichlet problem. By (1.28a), 2:rt

2~

2:rt

5U(P(p ei'f))dq>= 5U(F(p eit))dcp. 2

D

·

2~

1

0

We now consider the difference S(w) - U(w). This is also a subharmonic function in dP 1 with zero boundary values on CP 1 . By the maximum principle for subharmonic functions we have S(w) ~ U(w) throughout the domain dP 1 , with equality at some point w E dP 1 only if S(w) and U(w) are identically equal in dP 1 • Together with the last equality, this implies (1.28b), and equality holds only if S(w) is harmonic in dP 1 • As for the convexity of H(p) as a function of ln p, this property for functions S(F(z )) subharmonic in an annulus is well known as the Riesz convexity theorem* (see, for example, Timan and Trofimov [1], p. 194). Finally, if the function S(w) is such that H'(p') = 0 for some p' E (1, p0'), then the convexity condition for a differentiable function H(p ),

1 < P2

< p' < Po,

and inequality (1.28) for 1 < p < p' make (1.28b) an identity for 1 < p 2 < p', and this in turn is possible only if the function S(w) is harmonic in d P 1 . Q.E.D. Special cases of Theorem 1.3 yield various area theorems convenient for applications; one of these is the following

*See the Editor's Supplement to Part I.

I. THE SYSTEM OF FUNCTIONS {An(z)}

16

Let F(z) E ~, and let Q(w) satisfy the following conditions: (a) It is analytic in the complement dPo (p 0 > 1), with the exception of COROLLARY.

finitely many points. (b) The function IQ(w)I is single-valued and continuous in dpo· Then

!~": ! [

2~

r

IQ (F(pe;•))i' dcp] ;a, 0.

(1.28c)

We set S(w) = IQ(w)l 2 • Since IQ(w)I is univalent and continuous in dPo' the function S(w) is subharmonic in dP 0 • By Theorem 1.3, the mean value 21t

H(p) = 2~

SI Q (P (pel!f>))/

2

dcp

0

is a convex function of ln p in (1, p0 ). Since H(p) is differentiable for nonexceptional values p E (1, p0 ), the derivative dH(p )/d ln p = pH'(p) is nondecreasing in (1, p0 ). In view of (1.28), we conclude that H'(p) tends to a nonnegative limit as p -+ 1, and this proves (1.28c). Apart from the theorems considered above, there is another, very general area theorem of Lebedev [2] ; its proof makes use of the area principle for the complement of a union of several nonoverlapping one-sheeted domains with respect to the extended plane.

§3. Orthonormality of the system {vnA:(z)} 1.2. The following inequality is valid for the system {An(z)} generated by any function F(z) E ~: LEMMA

! SS B0

N

oo

2

N

LXnA~(z) dC1= L. k L.Xnank n=I

k=1

n=t

2

N

~ L. Ix~ 12 ,

(1.29)

n=l

where {xn} (n = 1, ... , N; N ~ 1) is an arbitrary system of complex numbers. Equality holds for a nonzero sequence {xn} in (1.29) if and only if F(z) E ~Suppose given an arbitrary function F(z) E ~ and an arbitrary nonzero system of numbers {xn}f (inequality (1.29) is trivial for the zero system {xn}). Define the function N'

Q(w) =

L ! Xnff n (w),

n=I

where 1Fn(w) (n = 1, ... , N) are the Faber polynomials generated by the function F(z), given by (1.11). The function Q(w) is a polynomial of degree at most N, not a constant. Thus Q( w) is regular in the complement d P for any p > 1. The

§3. ORTHONORMALITY OF-THE SYSTEM {y'nA~(z)}

composite function Q(F(z)) is regular in the annulus 1 it admits the representation N

Q (F(z)) =

< lzl
t.

(1.40)

1 1

n=l

I: nlA;(z)l n=l

2~

I. THE SYSTEM OF FUNCTIONS {An(z)}

20

Equality holds in (1.38) for finite z E B 0 only for functions F(z) E ~We first prove the assertion for v = 0. Fix an arbitrary finite S E B 0 (for z = oo the assertion is obvious) and consider any branch of the function I Q (w) = ln w-F (~)

This function is regular in the domain dP, p = Isl, since F(n is a boundary point of dP. The Laurent expansion of the composite function Q(F(z)) in the annulus 1

< lzl < p will now be derived from the identity 1n

1

F (z)-F (~)

=

l

z-~

n F (z)-F (~)

+ ln

1

(1.41)

z-~ ·

The first term in (1.41) is a regular function in lzl > 1, and therefore admits an expansion in negative powers of z, given by (1.2), while the second term is regular in the disk lzl < p and admits the usual expansion in positive powers of z. Thus, the identity (1.41) is rewritten

L ! t-nzn -In ( ~

ln

F (z).!_F (~)

=

~

~)+~An(~) z-n,

n=l

(1.42)

n=l

and hence, by Theorem 1.1, we get inequality (1.39) and also the assertion concerning the equality sign. Using (1.39), we can verify that ~TAn(nz-n converges uniformly in both variables z and on compact subsets of B 0 • Indeed, for any closed subdomain B~ = {z: lzl;.,: p}, p > 1, we have, by (1.39),

s

(n---+oo),

and this is a sufficient condition for uniform convergence. Now let v ;.,, 1, fix a finite E B 0 , and consider the function

s

Q(w) =

[in w-~m Jt.

Setting l~I = P > 1, we see that the function Q(w) is regular on the complement dP and not identically constant. In fact, the above vth order partial derivative with respect to is a polynomial of exact degree v in i/(w - F(n), since the coefficient of the leading term in (v - 1)! [F'(nt, which is not zero since F(n is univalent. The composite function Q(F(z)) is regular in the annulus 1 < lz\ < p, and we shall determine its Laurent series from (1.41 ), differentiating the latter v times with respect to

s

s:

21

§3, ORTHOi.JRMALITY OF THE SYSTEM {y'nA~(z)}

[in

F (z)~F (~)] ~,•) =

[

ln f (z;=; (~) ] + [1n

z1

~v)

~ ] t>.

Since ~~ An(nz-n is uniformly convergent in both variables z and

t on compact

B0 ,

subsets of it follows from Weierstrass's theorem that equality (1.2) may be differentiated arbitrarily many times with respect to each variable. If we differentiate (1.2) v times with respect to t, the last identity finally yields the Laurent expansion of Q(F(z)):

£

[ ln F (z)~F (~)] ~v) =

z-n + [tn (- ~)

A~v) (~)

n=l

l

(v)

+

.

t!

[~-n]Mzn,

n=l

and moreover, by Theorem 1.1,

L nlA~v)(~)l ~L ! l[~-n]Ml2, 00

00

2

n=l

n=l

which can be brought by simple manipulations to the form (1.38).

Q.E.D.

If the system {An(z)} is generated by a function F(z) E ~ and the sequence of numbers {xn} satisfies the condition ~~lxn 12 /n < 00 , then i:~xnA~v)(z) (v = 0, 1, ... ) converges absolutely and unifonnly on compact subsets of n°. COROLLARY.

1.6. For any system {anCz)} generated by an odd function F(z) E ~. the following inequalities hold for every v = 0, 1, ... in the domain Bo: THEOREM

(1.43)

L a,

n=l

I

a

1I

2

I

2 n l-(z~)-11 2n I a~vJ (z) 2 ~ ------~ azvfif;v v

(1 .44) ~=z

The equ(l,lity sign holds in (1.43) and (1.44) for finite z E B 0 only for odd functions F(z) E I. We shall only indicate the points at which the proof of this theorem differs from that of Theorem 1.5. In order to deduce (1.43), we consider any finite

t E B0

and construct the function Q1(w)=

[in :=;(~J]t,

while to prove (1.44) we consider the function [

Q2(w)= ln

.. 1 (w-F(~))(w+F(~))

]('V) ~.

22

I. THE SYSTEM OF FUNCTIONS {An(z)}

Each of these functions is regular in the complement dP, p = lrl, since F(t) and -F(n are boundary points of dP. We then expand the functions Q 1 (F(z)) and Q2 (F(z)) in Laurent series in the annulus 1 < lzl < p, and the statement of the theorem follows from Theorem 1.1. It is clear that use of the area theorem also leads to other inequalities for the systems {An(z)} which may be of use for investigating univalent functions in the classes S and L. However, in order to avoid repetitions this will be done for a general finitely connected domain in Chapter 5, while here only such results are derived as will be needed in Chapter 3 to analyze the coefficients of univalent functions.

§4. Simplest applications of the systems {An(z)} Our aim in this section is to show how the systems {AnCz)} yield remarkably simple and elementary derivations of many results in the theory of univalent functions, concerning distortion for mappings by univalent functions of classes L and S, boundary growth of these functions, conditions for univalence, and so on. All applications of the systems {An(z)} concerning the coefficients of univalent functions are collected in the third chapter. THEOREM

1.7 (li>WNER [l]). Let F(z) EL. Then for any z

= r E Bo

1-,2 ~IF'(~) I~ l~r2 ,

For finite

t

(1.45)

the upper bound in (I .45) is attained only for functions of the form

.

F (z) =

z-~ 1-

( _)_ 1

zs

+ canst,

which map B 0 onto thew-plane slit along an arc of a circle about w = F(n; the lower bound is attained only for functions of type

'

P(z)=z+; z- 1 +const,

'

t

which map B 0 onto the plane slit along a segment of a ray issuing from the point F(n. Given a finite

t

E B 0 , we set z

=t

in (1.2). Then (I>

ln P' (t) =

-

~ An (t) t-n.

(1.46)

n=l

Using the Cauchy inequality and (1 .39), we deduce from (1.46) that 00

llnP'(t)\~ ~ IAn(~)~-nl n=l

(1.47)

§4. SIMPLEST APPLICATIONS

23

and this implies (1.45). For the lower bound to be attained in (1.45), equality must hold throughout (1.47) and moreover we must have lnlF'(nl = -lln F'(nl, which is possible if and only if

An(~)=~

,-n

(n= 1, 2, .. . ).

But it follows from (1.2) that for this system {An(z)}

z-~ I In F(z)-F(s)=ln l-(~z)- 1 and this implies that F(z) = z bound is analogous. THEOREM

+ tz- 1 /f + const.

'

The reasoning for the upper

1.8. Let f(z) ES. Then for any z, lzl

~1

= r < 1,

1-r zf'(z) 1~l+r 1+,~ f(z) --.::::1_,, r

(l+r)z

I

{1.48)

r

~ /(z) I~ (1-r)z ,

l-r

(1.49)

l+r

(l+r>• ~lrI~ • ·

(1.50)

Equality holds for z i= 0 in each of these inequalities only for functions of the form Ill

I,a(z) =

e E (0 , 2n) ,

z = ~ ne- 1 (n-l) 8zn, (I-e-i8z)2 .k,.

(1.51)

n=l

which map lzl

< 1 onto the plane with radial slit issuing from

the point

-t0 /4.

To prove (1.48), we consider a function f(z) ES and construct the function F(z) = l/f(z- 1 ) E ~ 0 ; we then have the identity z2f' (z2)

s2F' (s2)

= F (~ = and t = 1/z E B 0 •

f (z2)

{1.52)

~F; (~)

2)

F 2 (s) '

where F 2 (z) = ..jF(z 2 ) E ~o This identity shows that to prove (1.48) we must find bounds for tF;(n/F2 (n. To do this, we evaluate ln(tF;(n/F2 (n), setting z =tin the expansion (1.5b), i.e. In

sF; (~) = F

-2

2 ({)

~ a

.k,. n=l

2n-1

(1") 1--c2n-11 '=' '=' •

(1.53)

Applying Cauchy's inequality and (1.43) for v = 0, we conclude from (1.53) that

I In ~

it~) I~ i: I 2

n=l

a 2 n-d~) ~-

1, r

I~

1-r• ~ F 2 (z 2)-F2 (z 1) z2 +z1 1+r 2 1+r2 -..:::: F2 (z 2 )+F 2 (z 1 ) z2 -z 1 -..::::1-r 2

(1.57) (1.58)

'

The first of these inequalities is known as the distortion theorem for chords under univalent mappings (Goluzin [3]), and the second* as the distortion theorem for the central angle (Bazilevi~ [3] ). Some other distortion theorems in the theory of univalent functions, and also conditions for univalence, can be obtained as special cases of the theorems proved in Chapter 6 for univalent functions in a finitely connected domain.

*See the Editor's Supplement to Part I.

CHAPTER 2

TAYLOR COEFFICIENTS OF A COMPOSITE FUNCTION

Suppose that Q(w) is a function given by a formal Taylor series (2.1) while the function w

= w(z)

is defined by a formal expansion (2.2)

whose coefficients b 1 , b 2 , ••• are arbitrary complex numbers. The composite function r2(w(z)) is defined by formal substitution of the series (2.2) into the series (2.1): 00

Q(ro(z))= ~Qk[ro(z)]k=B 0 +B1 z+B 2 z 2 + ... , k=o

(2.3)

and for fixed nk (k = 0, 1, ... ) its coefficients Bn are functions of the n complex variables bl> . .. , bn:

For applications, we would like to have sharp majorants for IBn I and various means of the coefficients. Particularly useful for univalent functions f(z) ES and F(z) EL is the solution of this problem in the special case Q(w) ew, since the univalence of a function is equivalent to relatively simple restrictions on the coefficients of the functions

tJi)

In z

z-t

In F(z) _ F(r)

and

Chapter 2 is devoted to this problem. § 1. Inequalities for the coefficients of a composite function in the general case

nk

0 (k = 1, 2, ... ); from the set of functions w(z), we select one, w(z) say, whose Taylor series (2.2) has positive coefficients: Let

~

27

=

28

II. TAYLOR COEFFICIENTS OF A COMPOSITE FUNCTION

Cl)

ro (z)

:.:.-=

~

k=l

bkzk, bk > 0.

(2.5)

Then we have LEMMA

2.1. For arbitrary complex numbers a~ and a~ (k

= 1, ...

, n)

such that

akak =

(2.6)

~k ,

bk

and any positive p and q such that 1/p + 1/q = 1, the Taylor coefficients Bn(bl' ... , bn) (n = 1, 2, ... ) of the composite function satisfy the inequality

I Bn (b1, b2, .. •, bn) I ~ B!IP (bi I a~ IP, ... ' bn I a~ IP)B~/q (b1 I a~ lq, ... ' bn I a; IQ).

(2.7)

Equality holds if bk = lbk Ir/, 111 I = 1, and la~ IP = la~ lq = lbk I/bk.

~

~

= b kbk/bk = bkak, so

For the proof, we express bk as bk

that

00

ro (z) = ~ bkakzk. k== I

Then the coefficients Bn of the composite function may be written

8" (bia 1 , bi a;, , , , , b,,an) =

~a,,," U,,," (a 1 , a~,

•.. , an),

(2.8)

'V

where Un,v = Un,vCa 1 , ••• , an) are certain products of integral powers of the numbers a 1 , ••• , an with unit coefficients: n

U,,," =

n a:k k:1

(nk ~ 0),

n

~ knk = n,

(2.9)

k=l

while cx.n,v are nonnegative and independent of ak. For example,

B, = (Q1 b,) a,+ (Q 2 2b1 "3) a 1 a 3

+ (Qi:) a:

a~a +

a~.

+ (Q 3 3b~b2 ) 2 (Q,b~) An examination of (2.9) easily reveals the following properties of the numbers Un,v• which will be used below: For any complex a~ and a~ (k = 1, ... , n):

(2.10) For any real A.:

1u,,,"(a1, a2, ... ,an)l).=U,,,"(la1l\ la2l\ ... , lanl>..). For every t:

(2.11)

§1. INEQUALITIES IN THE GENERAL CASE

29

(2.12) If ak = 1, then Un vCI, ... , 1) = 1. We now decompos; each ak =bk/bk (k = 1, 2, ... , n) in accordance with (2.6) into the two factors a~ and a~ and, using property (2.10), we see from (2.8) that

Bn (b1 , b,., ... , bn) =~an. v Un, v (a 1, a 2 ,

••• ,

an)

'V

=~an," Un," (a~, ... , a~)· Un," (a~, ••• , a;). 'Y

Applying Holder's inequality for the given p and q to the right of this equality, we get

I Bn I~~ a!;~ I Un,v (a~,

... , a~)

'V

Ia!;i I Un,v (a;,

... , a;) I

(2.13)

~(~an," I un," (a;,

... a~) 1P)1'p

(; an," I un," (a;,

Now, using (2.11), we rewrite (2.13) thus:

I Bn I~(~ an,'I un,'I ( 1a;1P, ... , la~ IP))1/p X

(;an,'I un,'I (Ia~ lq, ... ' I a; lq))1/q

I

and the right-hand side coincides with that of (2.7) by virute of (2.8). The assertion concerning equality in (2.7) is easily verified by using the following identity, valid for any t: (2.14) Lemma 2.1 implies 2.1. For any p > 1, the Taylor coefficients Bn(b 1 , (n = 1, 2, ... ) satisfy the inequality THEOREM

.•• ,

b n)

lBn (b1, • • •, bn) I

~ B!-l/p (b1,

•,,,

bn) B!/p ( bl

I;: Ip• ... , bn I~: Ip).

(2.15)

Equality holds in (2.15) if

bk= b1ick

(k = 1, 2, ... , n),

(2.16)

where c is an arbitrary complex number; if nk =I= 0 and B n =I= 0, this condition is also necessary.

30

II. TAYLOR COEFFICIENTS OF A COMPOSITE FUNCTION

This follows from Lemma 2.1 with a~ = bk/bk and a~ = I (k = 1, ... , n). The identity {2.14) again enables us to show that if bk = bkck (k = 1, ... , n), the left-hand and right-hand sides of (2.15) are equal. Now if nk =I= 0 (k = 1, ... , n), then, re-examining the proof of {2.7), we observe that an,v > 0 for all v in {2.8), since then an,v involves only positive factors. But then, for equality to hold in {2.15) or, equivalently, for a~ = bk/bk = ak and a~ = 1 (k = 1, ... , n) to hold in {2.13), it is necessary and sufficient that Un vCa 1 , ••. , an)= canst for all v. Since the n~mbers Un , v always include an, an_ 1 a 1 , an_ 2 ai, ... , a7, it follows from this condition that

whence, since Bn =I= 0, we get {2.16), since when Bn =I= 0 we also have a 1 =I= 0. The above proof is due to Lebedev {Lebedev and Milin [2]) for .Q(w) = ew, w(z) = ln{l/(1 - z)) and p = 2. The next theorem is a useful corollary of Theorem 2.1; we first introduce the following notation: {iµ(z)}n is the coefficient of zn in the expansion of iµ(z) about z = 0, (n

= 1, 2, ... ),

2.2. If for some p > 1 the Taylor coefficients of the functions .Q(w) and w(z) satisfy the conditions THEOREM

{2.17)

then

(2.18)

When nk

;;i.

0, equality holds in {2.18) if bk= bkck

(k = 1, 2, ... )

where c is an arbitrary number such that (2.17) is valid; and when nk equality holds only in this case.

{2.19)

>0

31

§ 1. INEQUALITIES IN THE GENERAL CASE

By Theorem 2.1, for each n

~:r~

= 1, 2, ... and p > 1 we have

Bn ( bl

It: Ip· ... ' It Ip) . bn

{2.20)

Summing {2.20) from 1 to any N, we get N

~ n IP ~ I-Bp-1

n=i 8 n

N

_..

=::::::::

~ Bn ~

(-b

1

I

n=t

bi bi

-;;-

Ip

-bn I=bn Ip) •

, • • ·,

bn

But by {2.17) the function wp(z) is regular in the unit disk and bounded there:

I©p (z) I~ cr,

lzl~ 1,

while rl(w) is regular in the disk lw I< a. Hence the composite function

is regular in lz I < 1. Its Taylor coefficients are nonnegative since the coefficients nk and bk lbk/bk jP (k = I, 2, ...) are nonnegative; and the partial sums Bn are uniformly bounded, since

Li

Hence the series 00

~

~ n=o

B(-b I=- .,p, · · ., I Ip) n

. 1

bi b1

-b

n

bll

::-bn

converges, and by Abel's second theorem

It remains to let N--+ 00, and this proves inequality {2.18). For equality to hold in {2.18), it is necessary and sufficient that equality hold in {2.20) for each n = I, 2, ... , and for this in turn {by Theorem 2.1) it is sufficient that bk have the form (2.19). If nk > 0, this is also a necessary condition; if bk = 0 this assertion is trivial, while if at least one coefficient bN is not zero then all the coefficients

Bn ( bl

It Ip' ... ' bn I;:: r)

32

II. TAYLOR COEFFICIENTS OF A COMPOSITE FUNCTION

are positive for n = kN, and consequently the coefficients Bn = Bn(b 1 , • • • , bn) do not vanish for these values of n; thus, by Theorem 2.1, condition (2.18) is also necessary. (to Theorems 2.1 and 2.2). Inequalities (2.15) and (2.18) are also valid when p = 1. In this case formulas (2.16) and (2.19) furnish the values of the coefficients bk which make (2.15) and (2.18), respectively, equalities. REMARK

§ 2. Inequalities for the coefficients of a composite

exponential function*

We adopt the notation 00

(I)

(z) = ~ Akzk, k=l

(2.21)

00

cp (z) = exp {ro (z)} ,

=

~ Dkzk

(2.22)

k=o

and call ',O{z) a composite exponential function. If w(z) is regular in the unit disk, then ip(z) is regular and nonvanishing in lz I < 1. Of all such functions ',O{z), the most conspicuous as regards the simplicity of the relations between the Taylor coefficients is the binomial function with positive exponent A.: (I __1_,1, =

.t.

exp {A !z•} = .t, d• (A) z•.

For any function w(z) or, equivalently, for any sequence of coefficients {Ak}~ in (2.21), we shall characterize the deviation of the system of numbers {Ak/A.}1 from {1/k}1 for each n = 1, 2, ... and A.> 0 by the following quantities L\n(A.), 8n(A.) and 5: (A.):

(2.23)

We now define the following classes of functions: 5A is the class of functions w(z) whose coefficients satisfy the following condition for a given A.> 0: lim &~(A)= & n ➔ oo

*See the Editor's Supplement to Part I.

< oq;

(2.24)

33

§2. INEQUALITIES FOR A COMPOSITE EXPONENTIAL

and eu and e 0 "A are the classes of functions 1,0(z) = exp {w(z)} where w(z) Ea or 5x, respectively. In several theorems of this section it will be proved that sharp bounds for IDn(A 1 , .•• , An)I and various means of the coefficients Dk cari be obtained by replacing Dk by dkCA.), if the parameter A is so chosen that 5n(>..) = 0. For other choices of A the majorizing term in estimation of linear combinations of the coefficients acquires a factor exp {½M: (>..)}. However, one can immediately verify that this is not the case for all >.., and in this connection the properties of the binomial coefficients dn(A) play a major role. 2.2. Let {Ak}~ be an arbitrary sequence of coefficients in (2.21), generating a sequence {Dk}; by the formal expansion (2.22), and for any >.. > 0 LEMMA

denote

en ("A ) --

n

dn

i

(!,+

n

' " 'Dk 12 1) ~ dk (A) exp k=o

(n

0 o{A)

{ -dn (t.+l) ,.

L dn-v ("A) 11

v

("A) }

,•=l

= 1, 2, ... ),

(2.25)

= 1.

Then {0n(>..)} is a nonincreasing sequence of positive numbers: (2.26)

The equality 0n(>..)

= 1 holds for some n ~ 1 if and only if A.

Ak = k 11k

(k

= 1, 2, ..• ,

n),

I 11 I= 1.

(2.27)

We start with the identity

zcp' (z) =

z 0 and every n

;;;;i:

1, {2.34)

Equality holds in {2.34) if and only if IAk I = ';.../k (k

= 1, 2, ...

, n).

Inequality {2.34) follows from {2.33) and the definition (2.23). In fact, dn

(t+

1)

±

dn-v (A) /),.v (A)~

'V=l

J.dd:(A(~~t f>n (A)~ t-6! (A).

II. TAYLOR COEFFICIENTS OF A COMPOSITE FUNCTION

36

For equality to hold in (2.34), it must hold in the last two inequalities, and this is true if and only if ~ 1 01.)

= · · · = ~n(X) = 0, which immediately.

implies the required assertion. COROLLARY 4. For every n ;;;;i. 1 and any X > 0, n

~ k I Dk 12 k=l

< Sn("-)Ad;(t.. + l) ex.p {A 0,

n

n

k=l

k=o

L. k IDk ! < t..dn-1 (A+ l} L. ~~(~;• < Adn (A+ l) L 2

k=l

and hence, by (2.34), we get {2.35). Note the useful fact that as n ~

"2:-7 klDk 12 , as

00

ld~(l; ,

the order of growth of the quantity

defined by the right-hand side of (2.35), is sharp for every X > 0,

since for the sequence Ak = Xr,k /k, Ir, I = 1, we have n

I.k!Dkl 2 " ' ; AdA(A+l),...,,

! r (:+l)n 2

2~

/?=I

(we have used the asymptotic equality dn(X

+ 1) ~ n">--/r(X +

1)).

Although Corollary 2 of Lemma 2.2 yields a sharp bound for any partial sum of the series "2:-;IDk 12 /dk(X), it does not easily furnish a bound for the sum of this series, and we must therefore resort to Theorem 2.2. THEOREM 2.3. If the Taylor coefficients Akin (2.21) satisfy the condition

for p > 1, then the coefficients Dk of the expansion (2.22) satisfy the following inequality for any X > 0:

"2:-;'= 1 kP- 1 IAk iP
.) < exp "'p-1

L. kP

k=o

k=t

k

(X)

-1

I Ak IP

}

•

(2.36)

Equality holds if and only if Ak = Xck /k (k = 1, 2, ... ), where c is an arbitrary complex number of modulus less than 1. To apply Theorem 2.2, we set !1(w) = ew and w(z) = X ln(l/(1 - z)). Then

~

~

in the notation of Theorem 2.2 we have bk = X/k and Bk = dk(X) (k = 1, 2, ... ). It is clear that the assumptions of Theorem 2.2 are satisfied, for

§2. INEQUALITIES FOR A COMPOSITE EXPONENTIAL

a,-,b

I

P

O'=Lbk bk k=l

=

37

a,

l

')..P-1Lkp-1IAklP 0, the Taylor coefficients of the expansion {2.22) satisfy the inequality THEOREM

IDn I< 0~1.:

1

(A) dn (A) exp { 2dn\A)

:t

dn-k (A-1) L\k (11,)} •

k=l

·

{2.37)

·

Equality holds for the coefficients {2.27). To prove the theorem, we apply {2.29) with k

= n,

at the same time using

the representation {2.33) for L~ - 1 1nk 12 /d k(X) (setting L~,: equal to zero for n = 1). The result is

f

dn _v(X)~v(X)

n

~ dn-k (A) I kAk \2 IDn 12~

k=l

en_i(A) dn-1 (11,

n2

x exp{ dn-1 ~"'+ 1)

+ 1)

fl

dn-v-1 (A)

L\v (A)} ,

V=l

which by virtue of the obvious equality dn-l (A.+ 1) = ndn(X)/X can be rewritten n-1 n 2 I Dn 1 ~ en-1 (A) d'fi (A) exp { nd:~A) dn-k-1 (A) L\k (A)} (A) \kAk \2

L

k~/rz-k

nAdn (A)

k= 1

If we denote the last factor by x and again use the inequality x ~ ex- I we have n-1 n f'•\~/n-k-dA) Ak(t.) (A) I kAk 12 2 2

I Dn I ~ en-1 (11,) dn (A)

exp

't

ndn U\)

+

,

k~/n-k

nAdn p,)

Rearranging inside the exponent, we get {2.37). A direct check shows that this is an equality for the system of coefficients {2.27). Since the coefficients dn-k(X - 1) in the exponent are negative for X < 1 and k =I= n, inequality {2.37) effectively yields bounds for 1Dn(A 1 , ••• , An)I only for X ;;., 1. Because of their importance in the applications, we state two corollaries of Theorem 2.4 in the case X;;.. 1.

38

II. TAYLOR COEFFICIENTS OF A COMPOSITE FUNCTION

COROLLARY

1. For every n

~

I, (2.38)

The equality 1Dn1

= exp{½An(l)}

holds only for Ak

= r//k (k = 1, ... , n),

1111=1. The last assertion follows for k = 1, 2, ... , n - 1 directly from Lemma 2.2, and for k = n from the equality that then holds in (2.29). COROLLARY

2. For any X ~ 1 and every n

I Dn I~ 8;/..:1 (A) dll (A) exp

{

"":~:c,Y-)

()n (A)}

~

1,

~ dn (A) exp { ; 6-J; (A)} -(2.39)

The equality IDnl = dnC>..) exp{M;:{X)/2} holds only for the coefficients (2.27). To examine the case X > 1, we must first prove several propositions. 2.5. For every n ;;;;, 1 and any ;\ of the expansion (2.22) satisfy the inequality THEOREM

n

~ I Dk

k=o

> 0, the

Taylor coefficients

n 12

~ exp {11,6-J; (A}}~ d!(t.).

(2.40)

k=o

Equality holds in (2.40) only for the coefficients (2.27). We first observe that if;\ ~ 1 the statement of the theorem follows in its entirety from Corollary 2 to Theorem 2.4, in view of the fact that {5:(X)} is a nondecreasing sequence. But if ;\ < 1, we start with the identity

Now, for X < -1 we have dk(X) - dk+ 1 (X) > 0 (k = 0, 1, ... ), and hence in order to estimate ~~ 1Dkl 2 we can use inequality (2.34). We get

again using the fact that 5:(;\) is a nondecreasing sequence of nonnegative numbers and the inequality 0k(;\) inequality the identity

~

1 (k

=

1, 2, ... , n). Applying to this

· 39

§2. INEQUALITIES FOR A COMPOSITE EXPONENTIAL

dk (A+ 1) =

k

L dv (A)= (k+l)

:Hl (,.)

{2.41)

v=o

for the binomial coefficients, we see that this is indeed the required inequality. If equality holds in {2.40), then necessarily 0n(X) = 1, and by Lemma 2.2 this can hold only for the coefficients {2.27). That equality does indeed hold for this system of coefficients is verified by a direct check, using the fact that Dk= dk(X) and 8:(X) = 0. Q.E.D. To complete Theorem 2.5 it is natural to evaluate the sum of squares of the binomial coefficients or, more precisely, the principal part of this sum as n~oo. LEMMA

2.3. For every n;;.;,, 1 and any positive X *½the sum of squares

of the binomial coefficients admits the estimate

{2.42)

Using the Abel transform we obtain n-1

n-1

n-1

k-1

L di (t.) = dn-1 (A)L dk (A)+ L [dk-1 (A) ..a-dk (A)) L dy (),) k=o

k=O

k=l

v=O

whence "J..

n-1

n-1

k=O

k=O

L. di ("A)= nd~ ("A)+ (1-A) L dk (A) dk+1 (1v)

(we have used (2.41) and the relation dk-I (A) - dk(X) For any A > 0, {2.43) implies the inequality n-1

(2A-1)

L di (A)~ nd~ (A),

k=O

and if A > ½ the first bound in (2.42) now follows. Next, we set

in {2.43), and the result is

= (1

{2.43)

- X)dk-I (X)/k).

40

II. TAYLOR COEFFICIENTS OF A COMPOSITE FUNCTION

n

-1

-1

'AL d!('A)=nd,;(A)+(l-A) L d!(A)+(l-A) L dk(~;i1 (A) 2

k= I

k=O

k=O

n-1

= (1-A) L d! (A)-(1-'A)(t-· d,;(A-)) +nd~(A) k=O

whence n-1

(l -211,)

Li di (1-)

k=O

n-1

= (1 -'A) (1 -d~ (1-))-nd; (11,)-( 1 -1-)2 L

k==O

dk (A) dk+dA) k+ l

For O < X < ½, this implies the second bound in {2.42). Note that if X = ½ we have the estimate n-1

LdHA-) < a In n+b k=O

where a and b are constants; but this will not ue used in the sequel. We have now prepared all necessary material for the main result of this section. THEOREM

2.6. For every n

~

l and any X > 0, the Taylor coefficients

of the expansion {2.22) have the following bounds:

exp {; 8t ('A)} 2 y2,2'"- l exp

{,. 6n+ (11,) } 2

dn ('A), l

dn (A-), n

IDnl~

t..~1, 2.-"At)(

t [2,., (I ~t)-l] ) 111

•

1)/4X and use identity 'n

dn (1

use Lemma

- t)]. As a result, from (2.49) we obtain _

'>.6,t ("A)} dn (1

= (2X -

> ½, and

n

+ "At) dn ("A-At)= k=O ~ dk ("At) ~ dy ("A-"At-1) v=O = ~

dk("At}dy(A-"At-l)

k+vn ~ dk('}..t)dv("A-At-1) n

< k=O ~dk(i.-l)=dn(i.),

§2, INEQUALITIES FOR A COMPOSITE EXPONENTIAL

43

we get (2.44b ). (We have here used the fact that dv('A. - 'At - 1) is negative for k + v > n.) The case 'A = ½. We first find a simple majorant for the quotient dn(l + 'A.t)/('A.t)½: namely,

{~1 t+}

n

dn (1 +"-t)

('J..t)''•

and then set, for n

~

II (1+ "-t) k

exp

k=I

=


1 we get

~ I 12 ::::::::2 ..... 1 (

~k Yk

k=l

~. 1

..1'.iT+

sp• tn- 1

k=l

I

I

t-1 dt+ n -

p

p

p

) -1

•

Setting p 2 = exp {2x/(2n + l)}, 0 < x < 00 , and transforming variables in the integral by t = exp {2r/(2n + l)}, we get II

I (

~

1

2

~2 ~k+2n+1 k- l

sx

1 . x ____.,!_ d1:+2n+1 o e2n+ 1 -e 2n+ 1

e•- e2 n\

_i:_

whence, by the obvioµs inequality for positive r and x 1 __,:____ ,:_ e2n+l_e- 2n+l

we get the estimate

But

5 X

•

--

e2n+l -1

~

x

d1:> 2n+l

and

0

where C is Euler's constant, and therefore

·< 2n+l 2~ '

·

:+ln _.x_

l

e2n+ I -e

)

__x_ ' 2n+ 1

§ 1. INEQUALITIES FOR THE LOGARITHMIC COEFFICIENTS

i;k

I Y,[' ~

.t !

+!

[f,

55

1 dT-lnx-C].

Setting x = In 2 and performing a few simple computations, we get the required assertion. In connection with Theorem 3.1, we note that for the Koebe function f 0 (z) = z/(1 - e-,e z) 2 we have 'Yk = e-ike /k (k = 1, 2, ... ), and therefore L7kl'Ykl 2 = L71/k. Moreover, for every n ~ 2 we have the inequality n

I:= k Ii'k 12 > L=

sup f

n

(z) E S k

k

l

l

!,

which is proved in §3 of this chapter. Combined with the results of Chapter 2, Theorem 3.1 turns out to be extremely useful for proving various bounds for the coefficients of univalent functions. Of particular interest is the behavior of the sums n

v

LL kli'kl

2,

'V=l k=l

for by virtue of (2.37) the truth of the inequality n

'V

n,

'\I

LI: kli'kl ~LL! v=lk=I 2

v:Jk=l

would imply that of the Bieberbach conjecture.* Before stating the next theorem, which is a special case of a theorem of Lebedev [2], we agree to denote by a(l/l(z)) the area of the image of K 0 (or B 0 ) under the mapping by a function 1/J(z) which is regular in K 0 (or B 0 ). Similarly, a(G) will denote the area of the domain G. THEOREM

3.2. If the function f(z)

the domain K 0 , if F(z) = "(Z domain B 0 , and if moreover,

+

= !~ cnzn

is regular and univalent in L;akz-k is meromorphic and univalent in the

(3.7) then

(3.8) Equality holds if and only if the area of the complement of f(K 0 ) U F(B 0 ) with respect to the extended plane is equal to zero.

Adopting the abbreviations f(K 0 ) = G and f(B 0 ) = D, we first assume that the domains G and D..are bounded by closed analytic Jordan curves, and *See the Editor's Supplement to Part I.

III. COEFFICIENTS OF UNIVALENT FUNCTIONS

56

introduce polar coordinates (R, ) in the w-plane, with pole at the origin. Let p ~ 1 be an integer, and subject the w-plane to the transformation (R 1 IP, ). Under this transformation, the domains G and D go into domains GP and DP (G 1 = G, D 1 = D), which still satisfy the inclusion relation GP C CDP (p = 1, 2, ...) and the resulting equality for their areas · (3.9) To determine the areas a(Gp) and a(CDP)' we consider the following univalent functions with p-fold symmetry:

{1//(zP) = c;1, (z + a1zp+1 + a2z2p+1 + ... ), 'j/F(zP) = -y'lp (z + b1z-p+ 1 + b2 z- 2P+ 1 + ... ),

ak

= ak (p),

bk= bk (p).

It is clear from the geometry of the transformations that

CJ (O,) = CJ ('V f (zP)) = n Ic, !''• ( I + CJ (CD,.) =n I'I'\'!, ( I Now, for every p

-t,

?;

(np

+ 1) Ia. I').

(np-1) Ib.

I').

= 1, 2, ... , we have

But it is readily seen that

P'

i;., n!a.l'=~ CJ(V'c'.~ )= ~ W(In 1;'>)'1'J'c'.:T•

so that (3.10) may be rewritten

P [a (Op)-n I c1 l21P] = \ ci\ 21 P SSI (inf K.O ,

da,

;z))' \2 I'c:~ \

21 P da

Cl)

+np \ C1 \2/p

L \an 12.

(3.11)

n=l

We claim that as p ~ 00 the second term on the right of (3 .11) tends to zero. In fact, it follows from the assumptions of the theorem that f(z) is bounded in izl < 1, i.e. that lf(z)l < M for z EK°, and therefore a(Gp) < rr.M2 1P. But then, in view of {3.10), we get the estimate

. (p -➔ oo),

57

§ 1. INEQUALITIES FOR THE LOGARITHMIC COEFFICIENTS

whence it follows that 00

PL. lanl

2

(p-+oo).

=o(l)

n=l

Letting p ~ 00 in (3.11) and using Lewy's theorem (see, for example, Kolmogorov and Fomin [1], Chapter V, §5.4 (English Chapter VIII, §30.1)), we get

limp [CJ (Op)-n I c1 rip]= (1 (inf (z>).

P➔ •

Z

Similariy, we find that

limp[njyj 11P-a(CD )}=o(lnF(z)). P~•

I

p

Finally, we have

a [ C ( aP U D p)]

= ~

SS R ~

-i

~

dR d)+o(lnF?))+

SS lwl-

2

d ]~

t, xz

I I' ,

(3.15)

where {x n} ( n = 0, l , . . . , m; m ~ 0) is an arbitrary finite sequence of complex numbers and the coefficients an k are defined by the expansion In F (z)-F (~) +In (v-u) z-~. _ 2 1n [ (z;-u.

yF

F(~~-v + VF (?-v. F (~-u]

a,

L

=

ankz-k~-n

( aoo

(3.16)

= Inv 4 u) .

n, k=o

Equality holds in (3.15) for nonzero {xn} if and only if F(z) E

j5_

We set h(w) = v'(w - u)(w - v),and define polynomials \Jln(w) ~ w71-l · · · by the expansion

+

(3.17) We then form the functions Qo (w) = 2 ln (Q0 (v) = 0),

(V w-u+ V w-v)-ln (v-u) · m

*See the Editor's Supplement to Part I.

(3 .18)

59

§ 1. INEQUALITIES FOR THE LOGARITHMIC COEFFICIENTS

assuming that the sequence {xn}~ is not zero, since otherwise {3.15) is trivially true. We assume moreover that x 0 is real; this involves no loss of generality, since the case of complex x O follows from the real case. We claim that S{w) = (Re Q(w)] 2 is a continuous subharmonic function on the complement dPo' for any Po> 1. To prove this, as is clear from the form of S(w), it will suffice to show that S(w) is single-valued in dPo by tracing the changes in the above functions as w describes a closed contour r not passing through the points u and u. Thus, if the interior of r contains u but not u, all the functions making up S(w) only change sign. If the interior of r contains u but not u, then Q0 (w) changes to -Q0 {w) + 21ri, Q 1 (w) to -Q 1 (w), Q(w) to -Q(w) + 2rrx 0 i and Re Q(w) to - Re Q(w). If both points u and u lie inside r, then Q 0 (w) receives an increment 2rri and therefore Q(w) changes by a pure imaginary quantity 2rrx 0 i, so that the increment to S{w}, as in the other cases, is zero. Thus the assumptions of Theorem 1.3 are satisfied, and therefore

H' (p) = But if we sett=

< lzl


0

(I

< p < p,).

(3.19)

in {3.16) and compare (3.16) with {3.17) in the annulus 1

we get the expansion Q0 (P(z)) =

00

In z-

~

a0kz-k

k=o 00

h(P(z))'P'n(P(z))=z"-n ~

ankz-k

(n=l, 2, ... ),

k=o

whence

and {3.20)

=

where z pt/"' and bk = ~~xnank (k = 0, l, ... ). Using {3 .20) to evaluate the integral in {3 .19), we get

H'(p)=

! d: {t.l~~I' p••+ t,lb.l-P-,. +2(Re(x,lnp-b

0 )] 2 }

60

III. COEFFICIENTS OF UNIVALENT FUNCTJONS

which gives (3.15) when we let p tend to 1. No information concerning equality in (3 .15) can be deduced from Theorem 1.3, since the latter concerns a more general class of functions S(w). The proof of our statement may be found in Pommerenke [4] and is not presented here, since we shall not need it in the sequel. COROLLARY (GARABEDIAN AND SCHIFFER [2]). If F(z) E values u and v (u-::/= v) in lzl > 1, then

~

omits the

(m= 1, 2, ... )

(3.21)

for an arbitrary system of complex numbers {xk}f, real x 0 and coefficients ank as in (3.16). We denote the left- and right-hand sides of (3.21) by A and B, respectively, and the second term on the left of (3.15) by 2C. Then it is obvious that

A

~ C + R{i,

Ct a.,x.) x~] ~

C

+ B' ,{~, k

11 a• IJ' . .,x.

(3.22)

Hence by (3.15) we get A~ C + [B(B - 2C)] 1 / 2 ~ B, which is (3.21). REMARK. Garabedian and Schiffer [2] also prove the following proposition. Let F(z) E ~. let p be sufficiently large, and let the numbers u and v be so chosen that (1) F(z)-::/= v in lzl > 1, (2) all the zeros of the function (3.23) lie on lzl = 1, and (3) u -::/= v is a root of the equation _l 2ni

5

m

"='"""-,---

.

~ xzk .. JF(z)-vdz=O

,k,. I z l=P k=-m

k

V

F (z)-u z

·

(3.24)

Then inequality (3.21) holds. Moreover, if we assume instead of (3 .23) that for l < m all zeros of ~~r kzk lie on the unit circle and the x k for k = l + 1, ... , m are sufficiently small, and if u is one of the l roots of equation (3.24) with minimum modulus, then (3 .21) is again satisfied. Inequality (3.21) was used by Garabedian and Schiffer to solve the local Bieberbach problem for coefficients of odd index. For functions of class S which grow rapidly near the unit circle, Bazilevic

§ 1. INEQUALITIES FOR THE LOGARITHMIC COEFFICIENTS

61

[7] established an inequality which estimates the nearness of the logarithmic coefficients to those of the Koe be function. To prove this inequality, we need

3.1. If the function f(z) ES is distinct from f 0 (z) then for every ip E [O, 21r) the quantity LEMMA

= z/(1

- e-i0z)2,

If (rel~) I (1-r)'

(3.25)

r

is strictly decreasing with increasing r (O

< r < 1).

For the function f 0 (z), when ip = 0 this quantity is identically equal to 1, and when ip -=I= 0 it is again strictly decreasing. We start from the identity

a In / f (re1'f) I R zf' (z) e f (z) = r or

'

z

= relq,.

For the left-hand side of this identity we have the bound (1.48), which is sharp only for fe(z), so that, since f(z) -=I= f 0 (z),

I

I

a1 L l +r i'Jr n f (re fP) If (r:e'fP 2 ) f (l - r . )2 2

,]

=M(r2, f)(l-r2)2,

'2

62

III. COEFFICIENTS OF UNIVALENT FUNCTIONS

which proves the assertion. For f(z)

= f 0 (z) we have M(r, f)r- 1 (1 - r) 2

= 1.

2. For any f(z) ES, there exists a direction of maximum growth; i.e., there exists 1{)0 E [O, 21r) such that COROLLARY

Jim I/ (relcpo) I (1-r)2 = ex.

(3.27)

r➔ J

But if If) E [O, 21r) and

If) -=/:- 1{)0 ,

then

lim lf(re 1cp) I (1-r) 2 = 0.

(3.28)

r➔ l

*

We may assume that a 0, since otherwise (3.27) and (3.28) are trivial consequences of (3.26). Take an arbitrary sequence of numbers such that rn ~ 1 as n ~ 00 , and choose ipn E [O, 21r) so that lf(rn/n) I (l -r)2 ~ I f (r neiCJln)I ( l _ rn)2. r

.

(J _29 )

,,,

Let ip0 be a limit point of the sequence {l{)n}, say l{)n k ting n = nk ~ 00 in (3.29), we get

~ 1{)0

as

nk ~

00 •

Let-

lf(r;iq>o)I (1-r)2 ~ a,

and in view of the inequality lf(r/ 0,

then the logarithmic coefficients of f(z) satisfy 0D

L k IYk -

k=l

where t 0

= e-iv,o, 1{)0

! t! ! ! In_!_, 2

~

(3.30)

a

being defined by (3.27).

Given the function f(z) ES, we construct the function F(n = 1/f(z) E ~ 0 , t = z- 1 E B 0 , and let {An(t)} be the system generated by this function. For any t E B 0 we have the identity 0D

L.njAn(~)- !

n=l

c-nr

CID

0D

(3.31)

0D

=LnlAn(t)J -2 ReL An(t)t-n+L ~ \tl- n. 2

2

n=l

n=l

n=l

The first term on the right of (3.31) is estimated with the help of (1.39); by (1.2) the second term satisfies the identity 0D

- 2 Re ~ An (t) t-n = 2 ln IP' (t) I, n=l

and the third sum in (3.31) is ln (1 -:- lt1- 2 (3.31) that

r

1•

Consequently it follows from

(3.32) for any t in the exterior of the unit disk. Now let t = pt 0 , p > 1. For the right-hand side of (3.32) we have*

lzl'lf'(z)I

IF'(t)I _

r 2 M(r,

f')

2ln l-l~l-2 -21n(l-lzl2)lf(z)l2 ~2ln(l-r2)lf(z)l2'

where z limit

= rz 0 , z 0 = /)=4n~klYkl 2 r 2 k~2nlnM(r,

t ),

and this, together with the preceding inequality with h replaced by X, gives {3.36). Q.E.D. We can now state the principal result of this section, originally proved by another method by Hayman [3] f9r circumferentially mean p-valent functions. To avoid complicating the notation, we shall write xn ~ 0 (n ~ 00) instead of Iimn ➔ ooxn=

0.

THEOREM

3.5. For any X >¼and any function f(z) ES,

{ (' ~) )"'} n

dn(2t..)

.-o}exp{i[Aarg/(re11P 0 )-(n+A)cp 0 )} (n---+oo),

{3.40)

where a and 'Po are defined by {3.26) and (3.27), while rand n are related by 8

1-T=n•

m1

(r)

lf(r)

= -,- (1 -r) 2

I;,.

exp { iA. arg / (r)}

. (n---+ oo ),

and in view of (3.42) and (3.27) this implies (3.40). Now let ex

= 0.

By Lemma 3.2,

where the positive numbers t E (1 - 1/2X, 1 - 1/4X) and r E (0, 1) are arbitrary.

68

III. COEFFICIENTS OF UNIVALENT FUNCTIONS

We set 1 - r

= 1/n and let

n-+

in this inequality. Then, in view of {3.26)

00

and the asymptotic equality

dn (1

n2>-t

+ 21.t) ,_,

r (l +2>..t)

(n-+oo),

we obtain {3.40). Q.E.D. COROLLARY

1 (HAYMAN [3]). For any function f(z)

=

L~cnzn ES, the

limit lim

n ➔ oo

~ n

=cx~l .

{3.45)

exists, and there/ore lcn I ~ n for n. > Nf"

ct=

To prove this, we need only set A= 1 in Theorem 3.5 and remember that 1 only for the function/o(z) = L~ne-i(n-t)Ozn ES. COROLLARY

2. For any funz 2 k-t, i.e. 2n

2~

oo

51t(re

1'1l)ld1 2 ,:w-1,

O 12 -14221 !11) = / cA2 > /2 •

By Theorem 3.5 with A=½, we have

69

§2. ASYMPTOTICS

lim Sn= lim I c}t 12 = a,

n ➔ QO

(3.47)

n ➔ a:,

and hence, by Abel's second theorem, we get (3 .46). This corollary is also due to Hayman [ 1] . COROLLARY

3. For functions f(z) ES such that

J~m~ where 0(1)

(t. l'-t, !) k IY,

= 0(1; 'Y1 , ')'2 ,

••• )

a> 0, (3.48)

= In 0~I)¾; In ex,

is defined by (2.32).

Introducing the notation IX)

~ ,m zn ( f z(z) ) t/1 = ~ n+l , n=O

we conclude from (3.1) that

Setting A. = 1 in (2.25), we get the following identity for the Taylor coefficients 'Yk and 42>:

or n n-1

"

-

-¼-?; ~ (k Ii'k 12- !)= In

~

I

c110 1

2

_ke_=n_1 __ i(_l)_n_

(n = 1, 2, ... ).

But by (3.47) we have n

~

lim

k=I

I c1u 1

2

--- = et, n

which, in view of (2.32), justifies the corresponding limit passage in the above equality, so that (3.49) Next, for any n

~

1 we consider the identity

70

III. COEFFICIENTS OF UNIVALENT FUNCTIONS

n

n

~ k Ii'k /

2-

where t 0

~

n

n

! = ~ kj i'k- ! t! j +Re~ ( 2ykt!-!), 2

1

= e-icpo.

{3.50)

Noting that

!)rk = In If (e','Po,) I

Rei: ( 2ykt~ -

(1 -r) 2 ,

k=l

and using (3.27), we find that 00

Hm Re Li

r➔ 1

(2ykt~ - ; ) rk= ln 0, then its coefficients satisfy the inequality THEOREM

· · · is in

the subclass

(3.56) *See the Editor's Supplement to Part I.

72

III. COEFFICIENTS OF UNIVALENT FUNCTIONS

where A f and Bf are constants depending only on the first three coefficients in the expansion of f(z) in the neighborhood of the pole. Given a function f(z) E S*(a), we consider the function F 2 (n = [f(z 2 )] -½ E :r 0 , t = z- 1 E B 0 , and the system {an(t)} generated by F 2 (n; for any t E B 0 we have the identity cc,

~ (2n-l rl p2a;n-1 (t)-t-(2n-2) 12

n=1 co

m

n=1

n=l

=p 4 ~ (2n-1) I a;n_ 1 (6)l 2 -2p 2 Re~ (2n- l) a;n-i (t) t-< 2n- 2 >

(3.57)

0D

+ 11=1 ~ (2n- l) It 1-2, An estimate for the first sum on the right of (3.57) is obtained from (1.43) for V

= 1: 0D

.

p• n~/2n-1) I a;n-i (~) 12

~ (ll~;:)

1

2 ,

ltl=->t. r

(3.58)

The second term in (3.57) can be found from the expansion (1.5b). To this end we differentiate (1.5b) with respect to z and t, the series :r~a2 n-t (t)z-( 2 n-t) on the right being differentiated term by term. This is legitimate by Weierstrass's theorem, since by virtue of (1.43) the series is uniformly convergent in both z and r on compact subsets of the domain B 0 . As a result, we get GD

2

L (2n-. 1) a~n-. (t) z1

211

n=I

and, setting z

2p 2

t

= t, we see that

(2n-1) a;n-t (6) 6- =

p: 6 {F (6), 6} + p; ( l~\~;>] 2

2

11

--

1) ,

n=l

where {F2 (0,

(3.59)

n denotes the Schwarzian derivative {F (,.) t} z ,., ,

F; (~) )2·

= ( F;(~) ) ' - ~ ( F;(~)

f; (~)

2

In view of (3.58), (3.59) and the equality

~(2n-l)ltl-2= (l-r') I+r'. 1,.1=..!_>1 r '

~

2 '

,.,

n=1

we deduce from (3.57) the following inequality for any

t E B0 :

73

§ 2. ASYMPTOTICS 00

L (2n-1) I p2a;n-i (~)-t-j 2 (3.60)

~ 2 (ll~:~2

Re[ ~2 ~• {P2 (t), ~} + ~2 (

~;;~~>)2-~

2 ].

Since f(z) E S*(a), there exists t 1 , It 1 I = 1, such that F 2 (t 1 ) = 0. To simplify the limit passage in (3.60) as t -+ t 1 , we shall assume that t 1 = 1 and use the following expansion of the function f 2 (z) = Jif}i) = I":~ b2 n-l z 2n-l irl the neighborhood of the pole z = I : /2

Vaz

(z) = l-z2 (1

+B

1

(1-z 2 )

+B

2

(t -z 2 ) 2

+ ... ].

(3.61)

As Bazilevic has proved [ 6] , the constant B 1 = bi is pure imaginary. Let us determine the right-hand side of (3.60) for t = p = 1/r > 1. Using (3.61), we see that as r--+ I

Re 6~ ~2 {P2 (~), ~}=Re 6~ z 2 { /2 (z), z} = 41 - 4 Re B 2 + o (1 ), Re

p2 ( tF; (t) )2 = Re pl ( zf; (i) )2 =-1_(1 +r2)' -3b2_4 R B + 2 4

F 2 (t)

4

f 2 (z)

4r

(l

- r')2

e

2

°(1) ,

whence 01)

~ (2n-1) I p2a;n_ 1(p)-r 2n- 2 j2 ~ 3b 2

+ 8 Re B + o (1 ). 2

(3.62)

We now observe that the relation (1.12) between the functions an(n and the Faber polynomials generated by F 2 implies the following relation between their derivatives:

(n

and this readily implies the existence of the limiting values a~(l). In fact, since f(z) E S*(a), the limiting value F;(I) = 2/yo. also exists, and therefore it follows from the above equality that

.!..n 3F~ (0) ,;= t + a~ ( 1) r a

(n = 1, 2, ••. ).

(3 .63)

Letting p = ,- 1 -+ 1 in (3.62), we get the fundamental equality needed to prove (3.56): 01)

~ (2n-l)la;n_ 1 (1)-1 l2 ~B< oo, n=t

where B = 3b 2 + 8ReB 2 • Next, since f(z 2 ) = ff(z), we have

(3.64)

III. COEFFICIENTS OF UNIVALENT FUNCTIONS

74

(3.65) But the coefficients b 2 n-l can be expressed in terms of a~n-l (1), by virtue of the expansion

obtained from (1.1 O) by differentiating with respect to w. Indeed, setting w

=

0 in this identity, we get

which together with (3.63) gives

V"; [1 +a;n-i (1)]

b2n-i =

Substituting our expressions for the coefficients

(n= 1, 2, ... ). b2k-l

(k = 1, ... , n) into

(3.65), we get Cn=

~ [n+2 kt/;k-1(1)+ i1a~k-1(l)a;n-c2k-ll(1)]

(n=2,3, ... ),

whence

or n

n

I en I ~an+aL Ia;k- 1 (1)-11 +~Li I a;k-l (1)-11 k=l

2•

k=I

Finally, using the fundamental inequality (3.64) we obtain

I en I~ an+ a (

t' n

1

+: L

2~ (2k-1) I a;k_i(l ) - 1 1

n .

l

)112

2k- l 1

n

(2k-1) I a;k_i(l )-11 2

k=l

~an+ a VB ( 2

+ ; ln n )1 12 + ~ B,

which proves (3.56). It is readily seen that B1 and A 1 depend only on the first three coefficients of the expansion of f(z) in the neighborhood of the pole.

§3. Bounds on the coefficients The regular growth of the coefficients of univalent functions of class S was first discovered by Littlewood, who proved that lcn I < en. Littlewood's

75

§3. BOUNDS

idea was to reduce estimation of the coefficients to estimation of the mean modulus. This approach was used to improve Littlewood's result by Landau [1], Bazilevic [2], and Goluzin [4]; finally, Bazilevic [3] and (independently) Milin (Lebedev and Milin [1]) exhausted the possibilities of the method, obtaining an asymptotically sharp bound for the mean modulus and a corresponding bound for the coefficients*, lcnl < en/2 + const. Subsequently, Bazilevic [4) replaced the mean modulus by a better majorant for jcn I, in the form of a certain partial sum, and proved that

lim sup ~ < 1.304.

n ➔ ~f(z)eS

n

Using (1.39), we can sharpen the general bound for the coefficients in class S. TuEOREM

3.8.* The coefficients of the function f(z)

=

L~cnzn ES admit

the bound

I en I < 1.243n

(n = 2, 3, ... ).

(3.66)

Given a function f(n ES, we consider F(z) = 1/f(n E L 0 , z = 1/t E B 0 ; for arbitrary finite w, consider the function z/(F(z) - w) and its Taylor expansion about z

=

00 •

Denote Q0

F

(zt-w =

L Pn (w) z-n . .

(3.67)

n=O

It is clear that the coefficients Pn(w) (n = 0, 1, ... ) are polynomials in w of exact degree n, since by (1.10) they are related to the Faber polynomials by the equality

We set w = 0 in (3 .67) and compare the coefficients of z-n on both sides, to get our starting representation for the Taylor coefficients of f(z): Cn=Pn_ 1

(0)

(n=2, 3, ..• ).

Since F(z) does not vanish in B 0 , it follows that the interior of any level curve CP = {w: w = F(z), lzl = p}, p > 1, contains the point w = 0. Then, by the maximum modulus principle for analytic functions, it follows that for n = 2,

3, ...

I en I= IPn-l (0) I~ max I Pn-i (w) l = max I Pn-i (F (~)) 1wec0

*See the Editor's Supplement to Part I.

ltl=o

(3.68)

76

III. COEFFICIENTS OF UNIVALENT FUNCTIONS

We now take the exponential of both sides of {1.2), thereby defining a system of functions {Dn(n} as follows: F

(,;=~(~)=exp

{l

Multiplying this equality by z/(z F(z)~F(b) =

An(~) z-"}

=

1

D. (~) z-•.

(3.69)

n, we get

t,(~• it•m~-•)z-•

If we set w = F(n in (3.67), where t is an arbitrary finite number in B 0 , and recall that the Taylor expansion of a regular function is unique, we conclude from (3.67) and the last equality that

n-1

Pn-1(F(~))=~n-1 ~ Dk(~)~-k k=o

(n=2, 3, .. .. ),

and this, together with (3.68), leads to the inequality

In±1

I en I~ pn-1

max Dk(~) ~-k I ~l=P k=o

I

p > 1.

(3.70)

Disregarding the fact that the right-hand side of (3.70) is strictly decreasing for decreasing p, since the polynomial Pn(w) is not a constant for n ;,. I, we shall let p E {l, 00) be arbitrary and use Cauchy's inequality to estimate the modulus of the sum in (3.70). Then, for n = 2, 3, ... , (3.71) But it follows from (2.36) for p

t. I

D, ( ~)I' ,a;;

ex P {

t.

= 2 and

k I A.

'X

=

1 and from (1.39) that

m1·} ,a;; 1 1,. ,

1 I ~I=,:-> 1,

which in combination with (3.71) yields the estimate

V 1-r2n

I en I< ,n-1 (l-r2)' Setting ,:i,n

= e-x, 0 < x
0 exp {AcS} dn (2A),

4V2A 4

1,. _1 exp {11,cS} dn (2A), n

~}~ !

Ve { -1u -exp 2 4

l (3.73) 1',-..:... - 4'

--'

nl/2

±-!)1/2 ( 211. [ 2e/~~f1,.)] 112 exp {11,cS} k=~

.

,

0

0. Then Clunie [1] constructed a univalent function of class L such that Ian I > n'Y-l ('y = 0.02) for infinitely many n. Recently, Pommerenke [3) improved these results for bounded univalent functions and for the class L. We present these results together with the author's proofs. 3.12. For every m = 1, 2, ... , there exists a function f(z) = L;;'=oamv+lzmv+t with nonnegative coefficients, regular and univalent in lzl < 1, lf(z)I < 1, such that THEOREM

I a,, I > no.1 7-1 for infinitely many n

(3.94)

= mv + 1.

We introduce a function h(z), regular in lzl a) Re h(z) > 0, lzl < 1; b) h(z) = 1 + L~cnzn ► 0; c) T.~cn/n = }\ < oo.

< 1, such that (3.95)

III. COEFFICIENTS OF UNIVALENT FUNCTIONS

96

(The notation L;=Oxnzn ► L;=oYnZn means that xn ~ Yn for all n = 0, 1, ....) Then, for any k = 0, 1, ... and any integer q ~ 3, the function A.

1. Leeman [Al] has established the sharp bounds _ 1090 ~ ~ 1090 1083...,.. b 7 ...,.. 1083 for all odd functions of class S with real coefficients. Bazilevic [ 1] applied the result of Fekete and Szego (in more general form) to obtain the sharp estimate lo: 3 1 ~ ½ + e- 6 for odd functions of class L. Much later, Garabedian and Schiffer [Al] used a variational method to establish this inequality in the full class L. Jenkins [A3], [A4] gave another derivation of this deeper result, based on the "general coefficient theorem". Bombieri [A2] found an alternate approach which simplifies the global analysis of the trajectory structure. Pommerenke [Al] found a different proof based on the inequality of Garabedian and Schiffer [2]. Grinspan [Al] gave an explicit argument to show that Milin's conjectured inequality (stated just before Theorem 3.2) on the logarithmic coefficients 'Yk implies Robertson's conjecture (Robertson [Al]) for odd functions, which implies the Bieberbach conjecture. Friedland [Al] used the Grunsky inequalities to prove Robertson's conjecture for the first four coefficients; see also Pommerenke [Al], p. 70. Aharonov [A2] and Nehari [Al] gave a different proof of Theorem 3.13. For further information on Bieberbach-Eilenberg functions and their generalizations, consult Aharonov [Al], [A6], Hummel [Al], Hummel and Schiffer [1], Jenkins [Al], [A2], [AS], [A6], Kuhnau [Al], and Pommerenke [Al]. Generalizations of the Garabedian-Schiffer inequalities have been given by Nehari [ 1] and by Schiffer and Schmidt [Al]. The proof of Theorem 3.4 (Bazilevic's inequality) can be simplified by applying the elementary inequality (1.48) instead of (3.33). Duren [A3] has

103

EDITOR'S SUPPLEMENT TO PART I

improved Theorem 3.7 by eliminating the logarithmic term. Duren [A2] also obtained a related result by applying a Tauberian remainder theorem. The mean modulus problem discussed in the introduction to §3 has been solved by Baernstein [Al]. More generally, he has shown that for each p (0 < p < 00) and for each r (O < r < I), the integral mean

Mp(r,

n = { ;Tr I:1T lf(ri0 )1P d0} l/p,

/Es.

is largest for the Koebe function k(z) = z(l - zr 2 and its rotations. The case p = I gives lcn I < en/2. It follows from results of Pommerenke [3] that Szego's conjecture (3.74) is false for all p ;;.i: 12. See Pommerenke [Al], p. 138, problem 4. The improvement by Il'ina [I] of Theorem 3.10 appeals to Milin's estimate lcn I < 1.243n. An appeal instead to FitzGerald's estimate lcn I < y7 /6n gives the further improvement A < 4.18. See Pommerenke [Al], p. 81. Early evidence that the Koebe function is a local maximum for the nth coefficient problem was obtained by Duren and Schiffer [Al] through the second variation. They essentially reduced the problem to the positive definiteness of a certain quadratic form, which they verified for n ~ 12. Bombieri [Al] then proved the positive definiteness for all n. Pederson [A2] proved the equivalence of several topologies near the Koebe point.

Part II UNIVALENT FUNCTIONS IN A FINITELY CONNECTED DOMAIN CHAPTER 4

THE LAURENT SYSTEM OF FUNCTIONS § 1. Existence and uniqueness of the Laurent system

1. The system of function pairs {tn(z), 11n(z)}. Let B* be an arbitrary

finitely connected domain containing the point at infinity whose boundary does not contain degenerate continua. Koe be [1] proved that there exists a unique function j 0 (z) E ~(B*), a 0 = 0, which maps the domain B* onto the plane slit along segments inclined at an angle 0 (0

< 0 < 1r) to the real axis.

Grunsky [1] generalized this result, proving that for every polynomial P(z) of degree n (n

=

1, 2, ... ) there exists a unique function of the form P(z)

f(z), where f(z) is regular in B*, f( 00)

r (z) 11/n ~ p,

=

1, 2, ... )

(4.35)

uniformly on every bounded subset' of the closed region B. For v = 1, this estimate and its uniformity follow directly from (4.33). For fixed v > 1, the proof is as follows. The fact that the polynomials {P~(z)} are uniformly bounded on compact subsets of b implies that their vth derivatives are also uniformly bounded on compact subsets of b, and this is sufficient to establish an inequality analogous to (4.33), yielding the desired result. Before stating the next theorem, we need some more notation. It was stated above that the Green's function for a domain B with pole at infinity has exactly m - 1 critical points (counting multiplicity) in the interior of B. Let these be the points z 1 , . . . , zm-l if the domain is multiply connected; if B is simply connected there are no critical points in B. Through each critical point zv passes a level curve of the Green's function, with parameter p(zv). If B is multiply connected, we let p0 denote the greatest of the numbers p(zv); if B is simply connected we set Po = 1:

Po={

max

p (z.,,),

1 Po then rR O is a simple closed analytic curve. In either case z = 0 lies either on or within the level curve r RO , i.e. in the closure bR O • Finally, in either case the closure bR O is a connected set. THEOREM

4.6. For any finite z, the polynomials Pn(z) admit the estimate

zEBRo, ~, zE bR0 , where E is an arbitrary positive number, M depends only on by (4.36). By (4.33), the derivatives P~(z) (n equality on the level curve rR O:

=

E,

(4.37)

and R 0 is defined

1, 2, ... ) satisfy the following in-

IP~(z)l~M(e)(l +e)nR~,

and by the maximum modulus principle this is also true in the interior of rRo· Since all the polynomials Pn(z) vanish at z = 0 and this point lies in the connected set bR O , in which the above inequality holds, it follows by integration that

To prove the second part of (4.37), we apply Lemma 4.2 to the polynomial Pn(z) in the domain BR 0 , on whose boundary rRo the lower inequality of (4.37) holds.

n

3. Properties of the domain functions R(z, and P(z, f). We first recall the definitions of the functions R(z, and P(z, f). For every fixed t E B (t =I= 00), the functions R(z, and P(z, f) are defined in B by (18); the functions j 0 (z, and i1r; 2 (z, in (18) are in class °L(B), vanish at the point z = L i.e.

n

n

n

n

(4.38) and map the domain B univalently onto the plane slit along radial segments and onto the plane slit along arcs of concentric circles about the origin, respectively. For t = 00 and z EB, we have P(z, 00) = R(z, 00) = 0. We now collect the properties of the functions R(z, and P(z, f) that will be needed later.

n

n

For fzxed t EB, the functions R(z, and P(z, f) are regular in z in the closed region B; for fixed z EB they are regular in t and f, respectively, in B. In the neighborhood of the point at infinity they possess Taylor expansions PROPERTY 1.

IV. THE LAURENT SYSTEM OF FUNCTIONS

122

IX)

00

R(z, ~)= ~ rn(~)z-n,

~

-

1 -

P (z, ~) -= ~ --,;:

n=I

~n

(z) z-n

n=l

and double-series expansions a-

R (z,

~)

I

L 'n bnk~-kz-n,

=

00

~-_IC r-kz-n ~ n nk"::, •

P(z, ~)=

n, k=l

n, k=I

Both functions satisfy symmetry relations: P(z, ~)

=

(4.39)

P(~, z).

This follows from the properties of the system {tn(z), 11n(z)}.

2. For every fixed tin the domain Br (l

1, then the sum of the series (4. 78) is a regular function on the bounding annular set Bl ,R and continuous on the boundary r of B. Moreover, the sum of the series (4.78) has a unique antiderivative on the set B 1 R• which is also continuous on r. COROLLARY

'

The regularity of the sum of (4. 78) on B 1 ,R and its continuity on r follow from the uniform convergence of the series on every closed annular set Bl ,p, 1 < p < R. Recall that for finite R the bounding annular set B 1 ,R is the union of as many domains as the level curve r R has components (i.e., at most m), while for R = 00 the set Bl ,R is the domain B punctured at z = 00• Since the series (4. 78) converges uniformly on compact subsets of each component of the bounding annular set B1 ,R, it is evident from the construction of the series that its sum has a single-valued antiderivative in these components, i.e. on the set Bl ,R. The continuity of the antiderivative on r follows from the continuity of the sum of (4. 78) on r. COROLLARY

2. Let A,, (n

= 1, 2,

. . . ) be any sequence of numbers such

that -

I

limjAnlI/n=-,

n->oo

R

and consider the corresponding series in terms of the conjugate system 00

~ A/I>n (z).

(4.79)

n=l

Suppose that (4.80)

142

IV. THE LAURENT SYSTEM OF FUNCTIONS

where R 0 is defined by (4.36). Then the series (4.79) is absolutely and uniformly convergent on every closed bounding set B 1 ,P, 1 < p < R. If R is finite, the series (4.79) cannot converge uniformly on any level curve rP, R < p < 00 • The absolute and uniform convergence of the series (4. 79) on every set B 1 ,P, 1 < p < R, follows from (4.52). If we assume that R is finite and (4. 79) converges uniformly on some level curve rP, p > R, it follows from Weierstrass's theorem that the series is also uniformly convergent on compact subsets of B 1 ,P, which implies that the series of derivatives (4.78) is uniformly convergent on level curves lying outside the annular set B 1 ,R; this contradicts Theorem 4.11. 4.12. Let {i\z} and {Xn} (n quences of numbers such that THEOREM

lim j An jt/n

n~~

=~ ,

= 1, 2, ... lim I An jl/n

n ➔ 00

) be any two se-

= r,

(4.81)

and consider the Laurent-type series 00

00

~ An©~ (z)

n=l

+n=l ~ Ancp~(z).

(4.82)

r 1,

Then the series (4.82) is absolutely and uniformly convergent on compact subsets of the annular set B,,R (if r < 1, we set B,,R = B 1 ,R) but nonuniformly convergent (if at all) on any level curve r P (p ·> I) lying outside B,,R. If either one of conditions (4.83) fails to hold, there is no annular subset of B, bounded by two level curves of the Green's function G(z, 00 ; B), within which the series (4.82) converges uniformly. All the assertions of the theorem follow at once by combination of Theorems 4.9 and 4.11. 2. Expansion in Laurent-type series. 4.3. Let~~ i\z~(z) be a series in terms of the derivatives of the conjugate system, whose coefficients satisfy the condition LEMMA

1 < 1. lim IAn ll/n = 00 R

n➔

(4.84)

Then for any function f(z) E a(B) (4.85)

143

§4. EXPANSION IN LAURENT-TYPE SERIES

We first assume that the function f(z) is continuous on the boundary r of B. In this case the integration in (4.85) can be carried out directly along r. But, by Theorem 4.11, since condition (4.84) is satisfied the series ~~ An ~ (z) converges uniformly on r, and therefore we may integrate term by term in (4.85). And since by Corollary 1 to Theorem 4.4 we have

~f

r

(z) 0 and p E (r, R) are arbitrary, this gives (4.90). But then, by Theorem 4.12, the series (4.92) converges absolutely and uniformly on compact subsets of B,,R. Consequently, the sum of this series is a regular function on B,,R, and moreover has there a single-valued antiderivative (this is clear from the construction of the series). We therefore denote the sum of (4.92) by l/l~(z): 00

00

~ Ak©k (z) + ~ Akcpk (z),

,q,: (z) =

k=l

z EB,. R·

k=l

Multiplying both sides of this equality by each of the functions 'Pn(z) and n (z) (n = 1, 2, . . . ), we integrate both of the resulting equalities along a level curve f' P' r < p < R. The right-hand side can be integrated term by term, subsequently applying biorthogonality on the boundary f' (formulas (4.28a-c)), while the left-hand side can be integrated by parts. The result is

An= 2~i

5

'\j,* (z) dcpn (z),

I'p

An= - 2~i

S'\j,* (z) d©n (z).

(4.93)

fp It follows from (4.89a), (4.89b) and (4.93) that for n

~

= 1, 2, ...

[ '\j, (z)- '\j,* (z)] cp~ (z) dz= 0, }

I'p

~

and p E (r, R)

(4.94) [ '\j, (z)- '\j,* (z)] ~ (z) dz= 0.

I'p

We now consider the difference l/l'(z) - 1/;:.,(z). This function is regular on

146

IV. THE LAURENT SYSTEM OF FUNCTIONS

Br,R, and its value at an arbitrary point

s E Br,R can be evaluated via Cauchy's

integral formula:

+-1 2ni

s r,,

,i, (z)- ,p* (z) dz (z- ~)2 . '

(4.95)

where R' and r' are chosen sufficiently near R and r, respectively, and r < r' < R'cp~(~), ~EB,,,

showing that the function 1/l'(z) - 1/l~(z) is analytically continuable from B , R'. r'

147

§4. EXPANSION IN LAURENT-TYPE SERIES

into Br·· But then the coefficients A~2 ) can be determined from the integral for-

mulas (4.64):

;n S[t

ii,~1> ~

(z)-11'* (z)] ~ (z) dz,

r,

and this, together with (4.94), gives 11.~2 )

= 0 (n = 1,

2, ... ), i.e. ij;'(z)

= iJ;'t,(z).

Thus, we have already obtained an expansion of the derivative 1/l'(z) as a Laurent-type series (4.88), converging absolutely and uniformly on compact subsets of Br,R• and also formulas (4.89a), (4.89b) and the inequalities (4.90) for the coefficients. The uniqueness of the expansion in Laurent-type series converging uniformly on compact subsets of Br,R follows from ( 4.93 ). If the function 1/l(z) is regular on Br,R (1

~ (z), n=I

n=I

where, by (4.90), the series L~ ~~(z) converges uniformly on every closed bounding annular set B 1 ,P' 1 < p < R, and its sum F'(z) (and also F(z)) is regular on B 1 ,R and continuous on the boundary r of B; the series L';'A,ill'~(z) converges uniformly on compact subsets of B, and its sum /(z) (and also f(z)) is regular in the entire domain B (Theorems 4.11 and 4.9). By (4.88), the following

equality holds on Bt ,R :

,t,' (z) = F' (z)

+/' (z).

(4.96)

We now assume that a*(i/1) is finite or, equivalently, that the double integral

~ ~ I,f (z) f• da, 1 < p < R. B

1, p

is convergent. Since the integral ff B 1 IF'(z)l 2 dais convergent (this follows ,P

from the continuity of F'(z) on f'), we conclude from {4.96) that the same is true of the double integral

S~ If' (z) 1 dn (z) + n=l

}:

n=I

An(j)n (z) + c,

(4.98)

which is absolutely and uniformly convergent on compact subsets of B, R. This ' expansion is unique; the coefficients ~ and A.n (n = 1, 2, . . . ) are given by (4.89a) and (4.89b), respectively, and satisfy (4.90). If l/l(z) is regular in B,,oo and admits a Laurent expansion of the form 00

\J, (z) =

}:

Cl;

Dnzn+Di} +}: dnz-n n=l

n=l

about z = 00, then the coefficients of (4.98) may be expressed in terms of D0 , Dn and dn (n = 1, 2, . . . ) by the formulas

) 00

An = }: kankDk, k=n

00

d i - ~ bk1Dk '\

_

_k_=_I_ _

1\,1 00

a 11

0

0

di-}:

bk1Dk

~

(4.99)

k=l 00

d 2 - } : bk 2 Dk k=l

00

ll1n ll2n • • • lln-1, n

dn- } : bknDk

An= _!__a_11_a_22_ _ _a_nn_ _ _ _k_=_l _ _~ (n = 2,3, . .. )-J We first observe that the annular set B,, R is indeed a domain, since when R > R 0 (for the definition of R 0 , see (4.36)) the level curve r R is a simple closed analytic curve whose interior contains the level curve r r·

150

IV. THE LAURENT SYSTEM OF FUNCTIONS

By Theorem 4.13, the derivative Vl'(z) admits an expansion in a Laurenttype series (4.88) which is uniformly convergent on compact subsets of Br,R. The coefficients of (4.88) are given by ( 4.89a) and (4.89b) and satisfy (4.90). Hence, first, by Corollary 2 to Theorem 4.11 the series ~.~ Ann(z) is absolutely and uniformly convergent on compact subsets of Br,R (it is even uniformly convergent on the boundary r of B) and, second, by Theorem 4.9 the series ~~An'Pn(z) is absolutely and uniformly convergent on compact subsets of Br and a fortiori on compact subsets of Br,R. It remains to fix a point z 0 E Br,R and integrate (4.88) from z 0 to an arbitrary point z E Br,R, and we then get the expansion ( 4.98). The uniqueness of this expansion follows from the uniqueness of (4.88). If V1(z) is regular in Br,oo• then, dividing both sides of (4.98) by z and integrating the resulting equality along a circle of sufficiently large radius, we determine the constant c, and formulas (4.99) then follow by the residue theorem from the integral representation of the coefficients An and An. 4. Examples of expansions.. EXAMPLE 1. Let t EB be fixed (t #: 00) , and consider the following functions of z: ,P 1 (z) = ln (z-~),

,P 2 (Z)=R(z, ~)-ln(z-~)=-; lnjn1 2 (z, ~)j0 (z, ~). Let us expand their derivatives in Laurent-type series on a bounding annular set.

4 If we set R = R(n = exp{G(t, 00 ; B)}, then it is clear that for fixed t EB any branch of the function VI 1 (z) is regular in the interior of the level curve rR and therefore on the bounding annular set B 1 ,R (R > 1) (see Figure 4). By FIGURE

Theorem 4.13, the derivative VI 1(z) can be expanded in a Laurent-type series (4.88) whose coefficients are given by (4.89a) and (4.89b ). Integrating by parts along a level curve rp, 1 < p < R, we see that for n = 1, 2, ...

151

§4. EXPANSION IN LAURENT-TYPE SERIES

Thus, the required expansion is ao

GO

z 1

~= L

-(J)n (~)

+L

, R > 1, not identically constant on any of the domains d~>. Suppose that the Laurent-type series expansion of the derivative of 1/l(z) = Q(F(z)) (1/l(z) is regular on the bounding annular set B 1 ,R), which exists by Theorem 4.13, has the form THEOREM

160

V. THE SYSTEM OF FUNCTIONS {Cn(z)} a,

a,

n =l

n=l

,t,' (z) = ~ An (z)

f ): (5.41)

166

V. THE SYSTEM OF FUNCTIONS {Cn(z)}

as follows from the expansion (4.19) of the function P(z,

f).

5.5. For any system {Cn(z)} generated by a function F(z) E 'J:-(B), an arbitrary system {xk} (k = 1, ... , N; N ~ 1) of numbers, given complex numbers t 1 , . . • , tN in the domain B, and fixed v = 0, 1, ... , we have the inequality THEOREM

(5.42)

~

Equality holds for every function F(z) E 'J:-(B). We may assume that all tk (k = 1, ... , N) are finite, for otherwise the equalities c~v)(oo) = IP~v)( 00) = 0 reduce everything to the proof of (5.42) with N replaced by N', where N' < N. We introduce the notation

R=-: min

(exp{O(~k' oo; B)}=--=cxp{O(F(~k), oo;

D)}),

R> 1, (5.43)

1-N

B)}

as

(6.8) For every fixed

r E Br, the function w = i/ln(z, n (n > N) is regular in z in the

closed domain Bn, and satisfies (6.8) on the boundary of that domain. In addition, for all n = 1, 2, ... we have

(6.9)

r

For every E Br, (6.8) determines m vertical stripes of width 2€, symmetric with respect to the straight lines Re w = -y~11 )(n (v = 1, ... , m), within which lie the images of the boundary curves r< 11 )(B(n)) under the mapping

n.

Let w = a be an arbitrary point not in the union of the above strips. By the argument principle this value is omitted by the function iJ; n(z, in the domain B(n), and so the image of B(n) lies in the interior of the m vertical

i/1 n(z,

n

n

strips. But the image of B(n) under iJ; n(z, is a connected set containing the point w = 0, and therefore in the closed domain B(n) (n > N) we have

I Re "Pn (z, ~)I< 2me,

~

EB,.

In view of (6.9) and (6.1), it follows from this inequality that the sequence of functions iJ; n(z, converges to zero uniformly in z and in the domain Br, i.e. uniformly on compact subsets of B. As mentioned before, analysis of the second inequality in (6.7) is analogous. Combining the results, we conclude that the sequences of functions R(z, B(n)) and P(z, f; B(n)) converge uniformly in z and on compact subsets of B, to the functions R(z, and P(z, respectively. By Weierstrass's theorem, a similar statement holds true for the sequences of partial derivatives of R(z, L B(n)) and P(z, f; B(n)) of arbitrary order. This proves the lemma.

n

r;

r

n

n,

r

The idea of the above proof is taken from Goluzin [1] . As our first application, we present a proof of the fundamental distortion theorem in the class 'E,(B), which is due to Grotzsch [1]. THEOREM

6.1. For any z

= r EB,

each function F(z) E 'E,(B) satisfies

the sharp estimate

(6.10) Equality holds for finite r only for functions F(z) = je(z, n + const, 0 ~ 0 < 1r. For fixed EB, inequality (6.10) defines the range of ln F'(n as a function-

r

al on the class 'E,(B).

· VI. APPLICATIONS

182

Consider an arbitrary function F(z) E L(B) and any finite t E B. Construct the function g(_z, by (5.8) and expand it in series in terms of the Taylor sys-

n

tem: 0,

Setting z

= t in this equality, we

get 00

lnF'(~)+R(~, t ) = - ~ en(~)cp 11 (~),

~EB1,011•

(6.11)

n=t

Estimating the modulus of the right-hand side in (6.11) by Cauchy's inequality and using Theorem 5.3, we obtain

I

,~1

en (t) cpn (t)

~

1~1

1-e,, (~) (f)n (~) I

= B and s is the extended plane punctured at m points. For fixed r E [r 1 , r2 ),

§ 1. ESTIMATES AND RANGES OF SOME FUNCTIONALS

183

the range .of the functional ln F'(n on the class ~(B(r)) is, as we have proved, a subset of the disk (6.15) which in turn is a subset of the disk (6.14), since 'J:,(B(r)) is a subclass of ~(B). As we have proved, the values of ln t; sPn(z) is uniformly convergent in z and t on compact subsets of B, we may differentiate term by term to evaluate the quantity on the left of (6.25). The result is 00 •

n

N

~

iJk~"

~ xkx"ozk 0.

(6.45)

rk

finite and distinct, then (6.45) necessarily holds (see Remark to Theorem 5.5). In this case, equality will hold in (6.38) if and only if equality holds throughout (6.43) for xk = yk and zk = rk; and this in turn is true if and only if N

N

~ Ykcn (~k) = e':!ifJ ~ Yk