Vector-Valued Functions and their Applications (Mathematics and its Applications: Chine Series 3)

Mathematics and Its Applications (Chinese Series)

Managing Editor:

M. HAZEWINKEL Centre/or Mathematics and Computer Science. Amsterdnm. The Netherlands

Volume 3

Vector-Valued Functions and their Applications by

Chuang-Gan Hu Department of Mathematics, Nankai University, Tianjin, People's Repuhlic of China

and

Chung-Chun Yang Department of Mathematics, The Hong Kong University of Science and Technology, Kowloon, Hong Kong

••

KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON

Library of Congress Cataloging-in-Publication Data

5a Chuang-g n. 193 . Vector-valued functlons and their applications I by Chuang-Gan Hu and Chung-Chun Yang. em. - - (Mathemat ics and its applicatlGns. Chinese series ; p.

Hu,

3)

Includes bibliographical referen c es (p. ) and Index. acid free paper) ISBN 0-7923-1605-3 (HB I. Yang. Chung-Chun, 19421. Vector valued functions. II. Title. III. Series Mathematics and Its ap�lications. series : 3. QA331.7.H82 1992 5 15' .9--dc20

Chinese

91-46130

ISBN 0-7923-1605-3

Published by Kluwer Academic Publishers, 00 AA Dordrecht, The Netherlands. P.O. Box 17,33 . •. • \! ..

,

'

Klu\'kr Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada

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© 1992 Kluwer Academic Publishers No pan of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

SERIES EDITOR'S PREFACE

'Et moi•...• si j'avait su comment en revcnir, je n'y semis point aUtI.' Jules Verne Tbe series is divergent; therefore we may be able to do something with il O. Hcaviside

One service mathematics bas rendered the human 1lICe. It bas put common sense back where it belongs. on the topmost shelf next to the dusty eanister labelled 'discarded nonsense' . ErieT.Bell

Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlinearities abound. Similarly. all kinds of parts of mathematics serve as tools for other parts and for other sci-

ences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered computer science ...• ; 'One service category theory has rendered mathematics ...•. All arguably true. And all statements obtainable this way form part of the raison d' etre of this series. This series. Mathematics and Its Applications. started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote "Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However. the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens. quite often in fact. that branches which were thought to be completely disparate are suddenly seen to be related. Further. the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma. coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as 'experimental mathematics'. 'cpo'. 'completely integrable systems'. 'chaos. synergetics and largescale order'. which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.•• By and large. all this still applies today. It is still true that at first sight mathematics seems rather fragmented and that to find, see. and exploit the deeper underlying interrelations more effort is needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will continue to try to make auch books available. If anything. the description I gave in 1977 is now an understatement. To the examples of interaction areas one should add string theory where Riemann surfaces. algebraic geometry. modular functions. knots. quantum field theory. Kac-Moody algebras. monstrous moonshine (and more) all come together. And to the examples of things which can be usefully applied let me add the topic 'finite geometry'; a combination of words which sounds like it might not even exist. let alone be applicable. And yet it is being applied: to lltatistics via designs. to radar/sonar detection arrays (via finite projective planes). and to bus connections of VLSI chips (via difference sets). There seems to be no part of (so-called pure) mathematics that is not .in immediate danger of being applied. And, accordingly. the applied mathematician needs to be aware of much more. Besides analysis and numerics, the ttaditional workhorses. he may need all kinds of combinatories. algebra, probability. and so on. In addition. the applied scientist needs to cope increasingly with the nonlinear world and the extra

VI

mathematical sophistication that this requires. For that is where the rewards are. Linear models are honest and a bit sad and depressing: proportional efforts and results. It is in the nonlinear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreciate what I am hinting at: if electronics were linear we would have no fun with transistors and computers; we would have no TV; in fact you would not be reading these lines. There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace and anticommuting integration, p-adic and ultrametric space. All three have applications in both electrical engineering and physics. Once, complex numbers were equally outlandish, but they frequently proved the shortest path between 'real' results. Similarly, the first two topics named have already provided a number of '·wormhole' paths. There is no telling where all this is leading - fortunately. Thus the original scope of the series, which for various (sound) reasons now comprises five subseries: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything else), still applies. It has been enlarged a bit to include books treating of the tools from one subdiscipline which are used in others. Thus the series still aims at books dealing with: a central concept which plays an important role in several different mathematical and/or scientific specialization areas; new applications of the results and ideas from one area of scientific endeavour into another; influences which the results, problems and concepts of one field of enquiry have, and have had, on the development of another. As the authors write in their preface, vector-valued functions tum up everywhere and they are a fundamental tool in physics, spectral theory, approximation and many other fields. Yet there was, so far, no systematic treatise on the topic. Here is one by two authors who have done quite a bit of wode in the field. At first sight one could think that a vector-valued function (instead of a scalar-valued one) would not bring so much new in the way of problems and theory. Just look at the components. It is far otherwise. There is a great deal more to vectorspaces, finite and infinite dimensional, then to scalars, and this takes an added emphasis in the case of vector-valued functions. To try to explain how much would take me far beyond the confines of an editorial preface; even if I could. It seems better to leave that to the authors, and the reader who wants to find out is therefore encouraged to pernse this unique volume. 1be shortest path between two truths in the real

domain passes through the complex domain.

J.Hadamard

Never lend books, for no one ever returns them; the only books I have in my bbrary are books that other folk have lent me. Anatole France

La physique ne nous donne pas seulement

I'occasion de ~udre des problMles ... eIIe

1be function of an expert is not to be more right

nons fait presseuIir Ia solution.

!ban other people, but to be wrong for more

H.Poincar6

sopbisticated reasons. David Butler

Bussum, 9 February 1992

TABLE OF CONTENTS

SERIES EDITOR'S PREFACE

v ri

PREFACE Chapter 1. Theory of Normal Families

1

1. PRELIMINARIES 1.1. The definitions of n(r. a). N(r. a). b(r,f). q(r,f). m(r. fl. mir,f) A(r. f). S(r. f) and T(r. f)

1

1.2.

Convex function

2

1.3.

1.4.

The properties of A(r,f) The properties of S(r. f)

3 5

1.5.

The properties of m(r,f' If)

10

1.6. 1.7.

The relationships among T(r,f). T(r. f') and N(r. fl. etc. The infinite product representations of a meromorphic function

16

22

2. THE NORMAL FAMILY OF MEROMORPHIC RJNCTIONS 2.1. The concept of normal family 2.2. The properties of the normal families of merom orphic functions

26 26 27

3. THE DISTANCE OF A FAMILY OF RJNCTIONS AT A POINT

38

4. ON MEROMORPHIC RJNCTIONS WITH DEFICIENT VALUES

42

5. THE APPLICATIONS OF THE THEORY OF NORMAL FAMILIES

53

6. APPLICATION TO UNIVALENT RJNCTIONS

60

viii Chapter 2. HP Space

68

1. HARMONIC 1.1. 1.2. 1.3. 1.4.

68 68

1.5.

AND SUBHARMONIC FUNCTIONS Harmonic Functions Boundary behaviors of Poisson-Stieltjes integrals Subharmonic functions The convexity theorem of Hardy Subordination

2. THE BASIC STRUCTURE OF lP'

71 73 76

78 79

2.1.

Boundary values

79

2.2. 2.3. 2.4.

Zeros The meaning of converging to the boundary values Canonical factorization

82 84 88

3.lP' IS A BANACH SPACE

3.1. 3.2.

Poisson integral and HI Banach space

Chapter 3. Vector-Valued Analysis

90 90 92 94

1. VECTOR-VALUED FUNCTIONS 94 1.1. Vector-valued bounded variation 94 1.2. Vector-valued integration 95 1.3. Vector-valued HOlder condition 100 1.4. Vector-valued regular functions 102 1.5. Compactness and convergence in the space of vector-valued regular functions 107 1.6. Boundary properties of a class of the vector-valued regular functions in S(O, 1) 110 1.7. Vector-valued elliptic functions 113

IX

2. VECTOR-VALUED BOUNDARY VALUE PROBLEMS 2.1. 2.2.

Vector-valued Cauchy type integrals Vector-valued singular integral equations

2.3. 2.4.

Vector-valued doubly-periodic Riemann boundary value problems Vector-valued boundary value problems with the boundary being a straight line

2.5. 2.6.

The solutions of the vector-valued disturbance problem The vector-valued boundary value problems in lp

3. THE ANALYSIS OF LOCALLY CONVEX SPACES 3.1. 3.2.

G-differentiability Vector-valued regular functions in locally convex spaces

117 117 119 123 127 133 137

145 145 148

mBLIOGRAPHY

151

INDEX OF SYMBOLS

154

INDEX

157

PREFACE The theory of vector-valued functions with one variable is one of the fundamental tools in studying modem physics, the spectral theory of operators, approximation of analytic operators, analytic mappings between vectors, and vector-valued functions of several variables. It seems that, thus far, no book specialized in dealing with vector-valued functions of one variable has been published in the West, except for some brief introduction to such functions in some books dealing with functional analysis or function theory. The present book will be a remedy for satisfying such a need. It consists of three chapters: theory of normal functions, If' space, and vector-valued functions with their applications, and it incorporates a lot of original research work obtained by the authors and others. The reader of this book is assumed to have a basic knowledge in real and complex analysis. The material of this book has been used as lecture notes for an optional course for senior mathematics and physics students at Nankai University. The approach of combining together function theory, functional analysis, and the theory of boundary value problems and integral equations has not only made this book unique, but has also benefited students in developing their ability in abstracting and generalizing mathematical theories. We would like to express our sincere thanks to Dr. DJ. Larner, the publisher of Kluwer Academic Publishers, for his interest and support in publishing this book. We'd like also to acknowledge the excellent typing job done by the Apparatus Office of Nankai University and Tianli Company, and to thank our students Mr. ZJ. Wang, Ms. D.P. Zhang and Mr. G.T. Ding for checking the final proof of this book. C.G. Hu C.C. Yang

CHAPTER

1

Theory of Normal Families

1. Preliminaries

1. 1. THE DEFINITIONS OF n(r,a), N(r,a), b(r,f), q(r,f), m(r,f) , mz(r,f),

A(r,f), S(r,f), AND T(r,f)

Let fez) be a function meromorphic in the disc Izl
=a

in I z I

< r;

a root of multiplicity k will be

counted k times. We set

N(r,a)=

J:

n(t,a)-;n(O,a)dt+n(O,a)log r.

In order to avoid confusion, when a-::pDO,we use n(r,l/(f-a»and N(r,l/(j-

a»

instead of n (r ,a) and N (r ,a) respectively, and when a =

and _V(r,f) are used for n(r,DO) and N(r,=) respectively. Set

1 q(r,f)=4:7r

J2" log (1 + If(re 0

08 )

IZ)dO.

=,

the notations n (r ,f)

CHAPTER

2

+ 1f(re 09 ) I) dO ,

fow log+ 0

mer ,f) = 211(

1

(log+x=max(log x,D)).

m2(r,f)=2~

J:" (log+ If(re09 ) 1)2dO.

1 A(r,f)=-;-

r'

pw

Jo Jo

S(r ,f) =

J:

1fl

(te 09 ) 12 If(te09) 12)2t

0+

df}

dt.

A(t;f) rIt.

T(r,f) =m(r,f) +N(r,f).

The properties and interrelationships among these functions will be discussed in the following.

1. 2. CONVEX FUNCTION

Let rp(r) be a real-valued continuous function defined in the interval [RJ ,R2 J and suppose that to any rj(j= 1,2,3) satisfying RJ'~,rJ
holds, then rp(r) is called a convex function on [RJ

,R2J.

Theorem 1. 1. 1. Let rp(r) be a convex function on (R J , R 2). Then to any positive numbers tj and points Xj' j= 1, ... n,

U!

.

(Rj>R 2 ).

~ tjrp(Xj) ~

j = J.

~tj j -

J

we have

rp( tJxJ +t2X2) ~ tJ rp(xJ) +t2rp(X2) • tJ +t2

tJ +t2•

The theorem follows by standard mathematical induction argument.

Theorem1. 1.2. l£t p,(r) and g(r) be tux) continuous functions definded on the cWsed interval

THEORY OF NORMAL FAMILIES

3

Assume that rp(t)

[ f:: r'

p(r)g(r)dr]

rp

J,:

p(r )dr

~

."""'"

f::

is a continuous convex June-

p(r )rp(g(r) )dr ]

'!

'.

p(r )dr

Proof. Let

r2- r l g. =g(rl+v--) (v=1,2,···,n).

n

o

Then, according to Theorem1. 1. 1, we have rp

(Phgh + ••• p.".g•• ) ~ Phrp(gh) + ••• + p.". rp (g •• ) Ph+···+p.". """'" Ph+···+p.". .

By the very definition of an integral, it follows that when n-oo,

.

and

r2- r l [~ ] - n - /='IPjorp(gjo) -

r'! p(r )rp(g(r) )dr. Jr!

The conclusion follows.

Proof.

The assertion follows immediately by virtue of Theorem 1. 1. 1 by letting

rp(t)=-log

t, p(r)=l, and g(r»O.

1. 3. THE PROPERTIES OF A(r,f)

Lemma'. ,.,. Let J (z) be

Junction in domain D, then .6IJ(z) 12 = 41J' (Z)12,

a reguJtrr

CHAPTER

4

6 where the ofK!T'aior

6

log 0

+

1

41f' (z)1 2

If(z)

12)= 0+ If(z) 12)2

(flu u re'Jll'esents the lAplacion: ax2

(flu

+ a y2·

Proof. Let f(z)=u(x,y)+v(x,y)i, where U,v are harmonic functions; hence 6u=O, 6v=O.

By a simple calculation, we have 6If(z)12

= 2[ (aU)2+ (dv)2+ (du)2+ (av)2+ u6u+v6v J ax

ax

ay

ay

=4If'(z)12

6 b

log 0

+ If(z) 12)

If(z)

12+

If

1261f 12_ [(~ If 12)2+ (~If 12)2J ax ay

But

(~If 12)2+ (~If 12)2 ax

ay

= 4(U 2+V2 ) [(aU)2+ (dy )2J ax

ax

= 4 If 12 If' 12. Hence

Theorem

'.'.3.

A(r,j)=b(r,j)+n(r,j).

Proof. First of all, f(z) has only finitely number of poles in the disk: Izl~r(r
Also let C 1 ,CH in:

Iz I~r

•••

,Ct be disks centered at these poles with sufficiently small radii all lying

and separated from each other.

Let Do be the region of Iz I
THEORY OF NORMAL FAMILIES

H..6.

5

+ If(te lll ) 12) dQ 2# (}log(1+ If(re lll ) 12)r dO- b fb (}logO+ If(z.) 12) P. dO., f log 0

DO

o

ar

Ib,I';;'

(}P.

0

where p,elll·=z.-b., z lies on the circumference of C. and

~

means the sum of the inte-

Ib,I';;,

Iz I~r.

grations along all the circies C. centered at b. that lie in

Letting P.- 0, the limit of the integration on the right hand side of O. 1. 1) tends to 4JTA(r,J). hl(z)

Let fez) be expressed as h2(z) , where hj(z) (j=1,2) are regular in the disk:

Izl
+ Ih 2(z.) 12)

/"2# (}log( Ihl(x.) 12 (}P.

Jo

-2

• dO P

f2. (}logl;2(Z.) I P. dO. o P.

Clearly, when P.-O, the first term of the right side of the above equation tends to zero. While the second term on the right side of the equation is equal to

-2 f2#(}loglh2(Z.) (}P.

o

I

df)

P.

= -4- • ..L /"2" () log Ih 2 (z.) 1ilJ) "

2JT

J

0

()

logP.

--4

•-

JTn.,

where n. is the order of the pole b.. Therefore, when P.- 0,

It follows that

A(r,f) =b(r ,f) +n(r ,00).

1. 4. THE PROPERTIES OF S(r,f). Theorem 1. 1. 4. Let fez) be a function merumurphic in Iz I~r, which is regular at point z= O.

CHAPTER

6

1

Then 1 S(r,J)=g(r,J)+N(r,oo)-ZlogO+ If(O) I)·

Proof. Clearly

JlogO+lf(te"')12)

.. has only a fInite number of poles on

at

Iz I =t(O
r). Let the moduli of these poles be denoted by tj (j = 1 ,2 , ••• , k) and assume O
r'j+,-"

f

2" J logO

41T

[logO

t

+ If I )2t dO)~+

at

o

=...!.. J2"

A(t,J) dt

J'j+'

t

+ If(te"') 12) Jt=tj+l-e' t=tj+e'

0

,J)+

=q(tj+,-e' ,J)-q(tj+e'

Pj+,-o' n(t,oo) dt J'j+" t ';+'-" n(t,oo) dt. 'j+r t dO+

J

Now letting e' -0 and by the continuity of q(r,J), we obtain

J,r'j+, -A(t,J) - t - dt=

q(tj+l,J) -q(tj,J)

J

+

r'j+,n(t,oo) dt

J

'j

(j=1,2, ••. ,k-l).

t

Clearly similar relations hold in (e,t,),"',(tpr); O<e
J: A(tt,J)dt

=

q(r,J)-q(e,J)+

=q(r,J)

J: n(t'too)dt

+ r' n(t,oo) -n(O, (0) dt J.

t

+n(O,oo)log r-[q(e,J)+n(O,oo)log

eJ.

Now by assumption that fez) is regular at z=O, it follows that n(O,oo)=O, and hence lim q(e,J) = 21 logO

•-0

+ If(O)

12 ) •

Theorem1. 1. 5. S(r,J) is a positive and monotonic increasing continuous functinn of r. Moreover , it is also a convex functinn of log r.

THEORY OF NORMAL FAMILIES

7

?roof The first part is obvious. As to the second part, the assertion follows by 0

dS(r,f) d log r

A(r,f).

and the fact that A(r,f) is a monotonic increasing function of r.

Theorem1. 1.

6. N(r,w) is a

'TnOIUJtonic mcreasing convex functiml of wg r. Moreover

N(r,w)-n(O,w)log r=

~

0< I_,(w)I";r

log

Ia. ~w ) I'

where a. (w) denotes a point of w- point of f with absolute vahle no greater than r.

?roof By conSlOdermg dN d (r log, w) r t h e fOlrst assertIOn f 0 11 ows. 0

0

0

Let rj be the absolute values of the w-points of fez) with 0
denote the number of w-points of f on Iz I = r j. Then rr n(t,w)-n(O,w) dt

Jot =.~ rrj+l n(t,w)-n(O,w) dt

.

.

J =

I

Jrj

t

= ~wg(i+l)·I+oo.+'j rj

j = I

~

w

r 9 la.(w) I'

0< 1_, (w) I";r

The theorem is thus proved.

Theorem1. 1.7. Let C(f)= {

-

21 logO

+ If(O) 12)

-log IC-.. I

(when 1(0) =00)

Then

Proof. Applying Theorem1. 1. 4 and by the definition of T(r,f) , we obtain S(r ,f) -T(r ,f) =q(r ,f) -mer ,f) +C(f).

But

CHAPTER

8

1

417f fo-log(1+ If(re iB ) 12)dO = 417f

fo- log+ (1 + If(re

~ 417f

J:-

12) dO

oB )

log+ I Cire oB ) 12 dO+ 417f

J:.

log2 dO

=m(r,00)+~2. On the other hand,

1..P" log + (1+ If(re oB I2) 47f Jo 1 ~-

47f

J2. log+ If(re

dO

oB ) 1 2dO=m(r,00).

0

Hence

Theorem1. 1. 8.

1 +7f2

(-2-)2A(r,logf(z))~A(r,f(z)).

Proof. First we note that to any k>O, the following inequality

1 ~7f2 (k+

i )~1 +

(log k)2+ 7f 2

holds. Hence

1 ~7f2 ( If(z) 1+ Iffz)

I )~1 +

(log If(z)

I )2+n2~1 +

llogf(z) 12,

consequently

f' (z) 1+n2 2 I~I ( - 2 - ) 0+ llogf(z)

If' (z)1 2 If(z) 12)2 •

12)2~0+

The assertion of the theorem follows from this and the definitions of A(T,log fez)), and A(T,f(z) ).

Theorem 1. 1. 9. Let f (z) be a mel'omarphi.c fUnction in the finite complex plane with f ( 0) #1 0,00 and If(O) I>e, If(O) l>e(e>O). Then there exists a constant K(e,j(O)) depending

on f ( 0) and e such that A(r ,log

f)~ (e+-!.Y A(r ,f) +~ {S(p,f) + K(e,f(O)) } e p-r

(r
THEORY OF NORMAL FAMILIES

9

(w=j(z))

=

ff

law 12 0+llogwI 2)2IwI 2

1.. 1';;. 1·1';;.

IT lawl 2 +lJ 10+llogwI2)2IwI2' .<1 .. 1<, 1,1';;.

+

ff

1.. 1;;'+

law 12 0+llogwI 2)2IwI 2'

(1. 1. 2)

1,1';;.

The second term of the right side of (1. 1. 2) is

ff

lawl 2 2)2IwI 2 10+llogwI .<1 .. 1<,

IT lawl 2 (lwl+~)2 =lJ 1 0+ Iw1 2)2 0+ Ilog w1 2)2 .<1 .. 1<, ~7I'(e+J.-)2 e

A(r,f).

And

where w. denotes, under fixed r, the point that maximizes the value of n(r,w) when Iw

I< e or Iwi> 1 / e. Thus

It follows that to any rand p satisfying r
Also by Theorem 1. 1. 4 N(p,w•.•

Furthermore, when

)~S(p,f)+ ~log( 1+ 1 ~~~.~~~; 1

Iw••• I ~e

2 ).

CHAPTER

10

1+w•.• f(O) 1

and, when

f(O)-w •.•

I

~ 1+elf(O) I If(O) I-e 1

-...:::

1 IW•• , I ~ -;1

1 +W•. rf(O) I ~ e+ If(O)

f(O)-w •.•

I

-...::: l-tif(O)I·

Now we choose [{= \

71UlX

{~lnn[I+(I+elf(0)1)2J ~lnn[I+( e+lf(O)1 2""!f

If(O)I-e' 2""!f

l-elf(O)1

)2J} •

the assertion of the theorem follows.

1. 5. THE PROPERTIES OF

m( rJ; )

Let fez) be a function merom orphic in Izl
fez) =co+c,zi+ •••

+c.z' + ... ,

co=f(O)#O,oo. It follows from the definition of A(r .I), that:

Let

Wg fez)

=WI

(z) =UI (z)

+iv l (z)

with -7rp>r

A(p,wl) +p+r 2(p-r) 2 :::>-: 1 -::- 27r(p-r) Thus by the corollary of Theorem 1. 1. 2 and the above, we have

Wg{ A(p,wl) +p+r} 2(p-r) 2

~Wg

{27r(;-r)

f:

1+IL12

fo- (I + I~d 2) dt dO}

1

THEORY OF NORMAL FAMILIES

11

(1. 1. 3) Also

(1. 1. 4) Since

it follows that

<_2_ p-r

<_2_ p-r

1 2 Jrr -2rr J2~ ZogO+rr2+ IU11 )dt dO p

0

1 J2~ JP -2 rr 0

{log2+Zog0+rr2)

r

4 =2log2(1+rr2)+-p-r

Now

Ul

= log If I,

JP -2 1 r

rr

+log + IU112}dt dB

J2n Zog+ IUlldt dB. 0

hence

21rr fow log+ IUl IdB <21rr J:~ log ( IUl 1+ 1) dB



21rr J:~ ( Iud + 1) dO


1 2rr

21rr fo" ~

J2W 0

(log+

1 If I +Zog+m+ 1) dB

[log ( IfI 2+1)+Zog(

=log{ q(t,f) +q( t,
~

1}12+1) }O} +log2

)} +Zog2

~ 0+ If(O) 1 )+ ~ log ( 1+ If(~) 12)} +2log2 2

(1. 1. 5)

CHAPTER

12

1

We note that S (r ,f) is a monotonic increasing function of r, hence

_4_ p-r

r 21 p

J.

1r

J2~ lng+

1Ul 1

dB dt

0

~4{ lng+S(p,f)+lng+lng 1~){ci~? 12 +3 lng2}.

(1.1.6)

It follows from (1. 1. 3) , (1. 1. 4) ,0. 1. 5), and (1. 1. 6) that

lng{ A(P,Wl) +p+r} +lng+l.. 2(p-r) 2 r

+ 21og2(1 +rr2) +4lng+S(p,f) + 12lng2 +

+41og log Thus by

1

+If(O) 1f (0)1 12 :;:::

2 p-r

f') dt.

Jp• q ( t , !

O. 1.

7)

Theorems 1. 1. 5 and 1. 1. 6 the monotonic increasing of S (r, f) and N (r ,f) ,

we obtain for any p>t>r

hence

O. 1. 8) By Theorem 1. 1. 9 (with the assumption that e< 1f (0) for any p' with R> p'

1,

e< 1f /0)

1'

e> 0) we have,

>p

1 )2 , A(P,Wl)~ ( e+- A(p,f)+-,p_{S(P' ,J)+[(e,f(O))}. e

p -p

Substituting

A(

p,

f)~~ ~p'-p

f(l p

A(t,f) dt

t

into the above inequality, we obtain 1 p' A(P,Wl )~-,-{ [(e+_)2+ I]S(P' ,f) +[(e,f(O))}. p -p e

0.1.9)

Hence

CD

Here Theorem 1.1. 4 and the fact 8(r.f)=8(r,L.(f»

rotation on the Riemann sphere.

are used, where L.(z) denotes transformation of

THEORY OF NORMAL FAMILIES

log +

1

2(p-r)

13

+log+p+r +2log2+log+~+log[(e+-.l)2 2 p' -p e

+WI] + S(p'

+ IJ

,1) +WI]+ K(e,f(O));;': WI]{ ~~:~~~ +Pt r } .

(1. 1. 10)

Combining (1. 1. 7),(1.1. 8), and (1. 1. 10),we get

WI]+-.l+WI]2 16 (1 +n2)2+2wl]+ p' +log+_I_+ r p'_p W +_I_+5w +S( I 1)+4w +w I] p-r I] p, I] I] WI] [ ( e++) 2

1+If(O) If(O) 12 +

+ 1J+WI] + K(e,f(O)) (1.1.11)

+2{ N( p,f; ) -N( r,f; )} >2 q ( rJ; ).

fl (-) Noting the poles of f(Z"') come from the poles and zeros of f and all are simple poles. We have

Let r
fl) -N (r f'lj ) ~N(p)-N(r). N-( P'j

(1. 1. 12)

Now assume rand p are given such that p' - r

p-r= S(p'

r

,1)+2

(1. 1. 13)

7T

Then, for r
p-r<S(pl,1)+2· According to Theorem 1. 1. 6 N (r) is a convex function of log r, hence N(p)-N(r)~N(pl )-N(r)~

log P-WI] r

log p' -wI] r

N(P') . log p' -wI] r

Since

log P-WI] r=log( I+ P r

r)~

p r r,

log p' _log r= -log( I-P' p' r);;,: p' p' r, We can derive from the above that

N(p)-N(r)~pl ; - r N(P' ). r

-r

CHAPTER

14

1

Th us by Theorem 1. 1. 4 and (1. 1. 13) we obtain N(p)-N(r) 2S(p' ,f) ++log(1 + If(O) 12) + <

~ log

1 1+ lf (0)12

S(P' ,f)+2

1 1+lf(0)1 2 <2+ 2 Iog If(O) I .

(1. 1. 14)

Also by(1. 1. 13) we have r ) p' - r p' -p=(p' -r) ( 1- p' [S(p' ,f)+2] ~ - 2 - '

and hence

1

Iog+--

p'-p

1 < Iog2+log+--. p'-r

(1. 1. 15)

Again by (1. 1. 13)

(1. 1. 16) substituting (1. 1. 12), (1. 1. 14), (1. 1. 15) and (1. 1. 16) into (1. 1. 11) and eliminating the term that contains p, and then replacing p' by p, we get q(r/; ) <3log+S(p,f) + log +

+ ;log+P+log+!

p~r + log e l 1(1 +7fl) + ~ log [( e++ 2

r

+ 1]

1 1+lf(0)12 1 +, +21og l\ (e,f(O)) +21og If(O) I +210 +log 1+ If(O) 12 9 If(O) I

1

When If(O) 1~1, choose e=2If(0)

I.

(1.1.17)

1

I'

and when If(O) 1<1, choose e=2 If(O)

Then whenlf(O) 1~1, ( e++) 2 <245 If (0) [(e,f(O) )
12 ,

+log If(O) I.

And when If(O) 1<1, (

1 )2

e+~

<

25 1 4 If(O)

12 '

THEORY OF NORMAL FAMILIES

15

, Iog10 1 A(e,f(O»~ -2-+ 109 If(O) I· Thus from(l. 1. 17) we arrive at the following theorem.

Theorem ,. ,. ,o.cNevanlinna). lPt fez) be a function meroJnorphi.c

co# 0,

U!

Izl
f(O)=

Then for R> p>r

(Xl.

m( r.t; ) >q( r.t; ) <3iog+S(p,f) 131 +Iog+--+-Iog+ p+iog+-+p( ICo I) +k, p-r 2 I' where

p( Ico I) =

~ ( iog+ leo I +Iog+ 1;0 I)

+~ (log+logl col +iog+iog+ 1;01)

O. 1. 18)

and

(1. 1. 19) For f(O)

= 0 or

(Xl,

i. e. fez) =zAg(z) , d~O, we have

m( r.t; ) ~m( ~ 1',

)

+m( 1', g; ) +log2

~iog+ Id I+iog+++m( 1', g; From g=

~, '"

) +log2.

(1. 1. 20)

we have

q(r,g)~ N(r,g)~

Hence by Theorem 1. 1. 4 for

q( r,~ ) +q(r,f), N(r,

d> 0 and,

~)N(r,f).

hence, for

d< 0 the following result holds: (1. 1. 21)

But

s(r, ~) = ~ (1+1'1241). Hence it follows from (1. 1. 21) , (1. 1. 22) and Theorem 1. 1. 10 that

(1. 1. 22)

CHAPTER

16

1

(1. 1. 23) When

d< 0,

we have

~logO+r2ol)
+ CIZ + ••• + c.z' + ••••

Then, when d~O with C-l#O

m( r ,L) < 3log + S(p,j) +log+_I_ f p-r 3 + p+2log + - 1 +-log 2

r

+3{ log+log+r+log+Zog+ ; } +4log+

Idl +p( IC-li )+kl ,

where p(

Ic.ll)= ~ { log+ Ic.ll +log+ Id-4l}

+ ~ { log + log + 1G. 1+log+log+ IC~ll} , 4

kl =k+log2 IO •

(1.1.24) 0.1.25)

1. 6. THE RELATIONSHIPS AMONG T(r,j),T(r,!,), AND N(r,j) ETC. Let f(z) be meromorphic in

Izl
C.l + C.(oI-J) C.I + Co + CIZ -;: +01-1 + ... + -;+"·+C.z'+··· (tl~O),

as its Laurent expansion around z= O.

THEORY OF NORMAL FAMILIES Let

WH W2 , ••• Wq

From f'

17

be q distinct finite values.

= f • f; ,

it follows that

T(r,f' )=m(r,f' )+N(r,f' )

<,m(r,n+m(r,~

(1. 1. 26)

)+N(r,!').

We shall use the property

T(r,n=T(r, to get an estimation of T(r ,f'

~)+logIC_II,

).

First of all, we have

T(r,f' )=T(r,} )

=N(r'f~

+ log IIL, I

)+m(r,} )+logIP-II,

(1.1.27)

where

P-I= {

-dc_I> when d:;i: 0 , Icc., when d=O

and c. is the first nonzero coefficient in the Laurent expansion expressed above. Put q

F(z)f' (z) '" 1 F(z)= f' () , F(z)=L.J f( Z )_ w. ' Z .=1 then

(1. 1. 28) or

1 [ 1+ '" F(z)= f(z)-w L.J f(Z)-W~J f(z)-w ' p.E {I,2,3,···,q}. ')"#14

I'

(1. 1. 29)

t'

Set

o(w) =min{

Iw,-wil :h:;i:k,

l<,h,k<,q}.

Then, for any z satisfying inequality

If(z)-w~ I
O. 1. 30)

CHAPTER

18

1

Hence

~

.!.=.!..

1 fez) - W p 1
3

3q

fez) -w.

'#P

Consequently 1

1+ ~ o#P

f(z)-w p f(z)-w.

1

>l... 3

Thus for any z satisfying O. 1. 30). from (1. 1. 29) we have wg+ IF(z)

Now the arcs on

Iz I = r

I>

wg+ 1 f(Z)l_ Wp I-W9 3.

that satisfy (1. 1. 30) for J.l = 1 • 2 •.•.• q are disjoint from each

other. Thus q

1lI(r.F)~ •

>~~ 2n ,,=1

1 ~ -2 n

p=1

J

6(w) If-w.I<""2q

J

wg+ IF(re 08 ) IdO

I

If-w.I<6~~) lng+ f(re 081 )-w" IdO-W g3.

0.1.31)

While

1 - 2n

J

I

I

IJ-w.I;;.6~~) lng+ f(re 081-wp dO

~m(r'f~wJ-Wg+ o~~')'

0.1.32)

Therefore. it follows from (1. 1. 31) and (1. 1. 32) that q

1lI(r.F»~1lI(r'f _I

Combining this with

and we obtain

1

~

)-q wg+ q)-Wg3. X(2

u

W

0.1.33)

THEORY OF NORMAL FAMILIES

19

q

-q log2-

~ log+ Iw.I-log IGo I,

0.1.34)

tt=l

q

where Go is the first nonzero coefficient of the Laurent expansion of G(z)

= II (J(z)1'=1

w,o) around

z= O.

From (1. 1. 27), (1. 1. 28), (1. 1. 33), and (1. 1. 31), we get

T(r,f')~N(r'f~ -

)+QT(r,f)

~N(r,_l ) -m(r, ~ -.L) f-w. f-w" v=1

._1

+log la-41-Q

2Q Iog+---log3-Q log2 6(w)

q

~log+ Iw.l-log IG o I· .=1

-

o. 1. 35)

Furthermore since

m(

r

~ f f' uJ =m( r,~ ) ,

and by Theorem 1. 1. 7, we obtain

1_+1..log+ p m(r ,~) <3log+S(p,G) +log+_ G p-r 2 +2Iog+ ;

+3{ 109+log+r+log+log++}

+ 4log + Imo I +p( IG o I )+k l

,

0.1.36)

Where

p( IGo I) =

+

H

; {log+ IGo I + log +

log+log+ IG o I

I~o I}

+log + log + I~o I} ,

kl =k+log2 IO ,

=, mo=O if G(O):;t:O,=. It follows from Theorem 1. 1. 7 (regardless of G (0) :;t: = or G (0) = = )

and modenote the multiplicity of G(O)=O or that

(1.1.37)

CHAPTER

20

1

Now as it is easily shown that T(r ,fg)<,T(r,f) +T(r ,g), T(r,f +g)<,T(r,f) +T(r,g) +log2. We have q

T(p,G)<, bT(p,f-w.), tI=J

T(p,f-w.)<' T(p,f)+l.og+

Iw.1 +log 2 , q

T(p,G)<,qT(p,f)+qlog2+ blog+ .=1

Iw.l·

Hence I.og+ S(p,G)<,2log q+log+T(p,f)

+ log + ( ~log+ Iw. l ) +2I.og3+l.og+l.og+ 1~o I·

(1.1.38)

Thus by combining(1.1. 26),(1.1. 35),(1.1. 36),(1.1. 37),(1.1. 38),and applying Theorem 1. 1. 11 to m (r , ~ ), we can obtain the following.

Theorem 1. 1. 12.

, qT(r,f)- bN(r,_I_) .=1 f-w

+log+_I_+1-log P+21.o9+J...+ 3 ( I.og+ I.og+r+log+log+J...) p-r 2 r r

,

+

,

~log+ Iw" I + 3log + ( ~log+ Iw" I)

+3log+log+ 1 +q I.og+ o~!)

~o 1 +log IGo I +p( IGo I) + 6log q

+ 4I.og + Imo I +7log3+q I.og2+k

l }

+logla_41<,T(r,f' )<,N(r,f' )+m(r,f) +3log+T(p,f)+3I.og+log+

Ic~ I+31.og3

1 3 1 +log+--+-log+ p+2log+-+ 3 {log+log+r p-r 2 r

THEORY OF NORMAL FAMILIES

21

From

S(r,f'

)+11I(r,f)~N(r,f'

)-2N(r,f)+2N(r,f)+2111(r,f) ,

and from the above theorem it follows that q

(q-2)T(r,f)~ ~N(r'f

1

tI=l

) -NI (r)+D(p,r), Wt!

)+N(r'f~

N 1(r)=2N(r,f)-N(r,f'

(1.1.39)

),

where

D(p,r) =6log+T(p,f)

+ 2log+_1_+ 31og+ p p-r

+4log+++6( log + log+r+ log + log + ; )

IC~l I+ 4log + 1111 I+p( IC-ll)

+3Iog+log+ q

q

_I

_I

~Iog+ Iw.1 +3log+( ~log+ Iw.,I)

+

+3log+log+

I~o I+log IG

o

I +p( IG o i)+6log

q

+qlog+ o~!) + 4log + Iilia 1+ 1 olog 3 +qlog2+2k)-log la-ll. Moreover, if, in Theorem 1. 1.12, by adding T(r,f)-N(r,J) and

0.1.40)

111(1' ,f) =111(1',

n+2S(r,f)-2N(r,f) to the left side and right side of T(r,f'), respectively, then by a simple calculation then q

(q-l)T(r,f)~ ~N( 1', f

1

w.) +N(r,f)-.N) (r)+D(p,r).

Thus the following important result is obtained. Theorem 1. 1. 13.( Littlewood- Collingwood- Nevanlinna theorem or Nevanlinna' s second fundamental theorem). LRt fez) be meranwrphic in

Iz IW2'"

,Wq

be q distinct val-

"Ues(posslhly rooy include (0). Then, to R> p>r>O, the following inequality holds: q

(q-2)T(r,f)<

~N( 1', f

1

wJ

-N1(r)+D(p,r).

CHAPTER

22

Hfffe, according to (1. 1. 40), when

WI , •••

1

,Wq all are finiJR, the term D(p, r) stands as it

is; when one of the Wi s, say Wq = 00, then in O. 1. 40) q - 1 will replace q m the upper mdex of the summatians toot cancfffn the vi s.

Corollary. Let f (z) be 7T1R1'Olnmphic in Iz I
Iz I
WI

,W2 , ••• ,Wq be q (q ~ 2) distinct vahws ( 00

then for R> p>r>O

(q-2)T(r,J)
1. 7. THE INFINITE PRODUCT REPERESENTATION OF A MEROMORPHIC FUNCTION

We shall consider the representation of a finite order merom orphic functions, in the plane, by a infinite product formed with their zeros and poles. The following result is standard.

Theorem 1. 1. 14. Let f(z) be a 7T1R1'o/nmphic functinn of ordfff ).. Then

(1. 1. 41)

whfffe p is the the smaUest integer(inctuding ZfffO)toot sattsflRs p+ 1>)" k is an mtegfff(negatwe or positwe) or ZfffO, P (z) is a polynornwZ of degree

<. p, IT

represents a product, withm the

disk: Iz I
The right side of expressIOn O. 1. 41) dO/Mill

til

uniformaly in generaZ sense ill allY bounded

D not cOTliainmg Zfff08 or poles of f (z ). CD

D --"-ly, tf o f r the'mteger p, r!~ Z· rart""u
CD

conl¥'1'geS

rr ~ T(t,J)dt ts . Jr.

f"" .~ mite, t,te,.

Converges uniformly in general sense In D means it converges uniformly in any closed subregion of D.

THEORY OF NORMAL FAMILIES

23

(1. 1. 42)

Proof. By assumption lim log T(1' ,f)

wg

r-=

=).,.

l'

It means that to any given e>O, there corresponds a 1'o(e) such that whenever 1'>1'o(e) ,

T(1',f)<1'H,.

Now p+ 1

>)."

so e can be chosen satisfying p+ 1>).,+e. Thus

ttm T(1',f)

O.

o

r_oo

Consider the case that f(O)#O or

00.

r ,+1

Then for f(z)#O,oo, by Poisson-Jensen formu-

la,we have

loglf(~) I~+z

log f(z)=J... f2n 2:rr 0

~-z

dO-

L;

log

1• .I';;p

a~z-p2

p(z-a~)

+ Ib.I';;p L; wg b"Z- p +ic (~=pe~). p(z-b.) 2

0.1.43)

Taking derivative with respect to z, we obtain from above d wgf(z)

dz

+ L;

_1__

IG.I';;;p z-a~

L;

_1_.

Ib.I';;;p

z-b.

0.1.44)

By taking the p-th derivative of the above, we have d,+1 dz'+ 1 log (z) -(-1)',

=

(p+ 1)! J2n 2~ 2:rr 0 log If(~) I (~_z)'+2dO

p!

(1. 1. 45) In order to derive the limit of the right side of (1. 1. 45) when p-oo, we denote

the right side of the above equation as =/ 1 +/ 2 +/3 ,

CHAPTER

24

First of all, for Iz I
1/11~(pt:)!

J:" lloglf(~) II (p!.;)'+2 dO ,

but 217r

fa" Ilog If(~) lidO

=1 - J2" log+ If(~)

27r

0

=m(p,f) +m( p,

~

1 IdO+27r

J2" tog+ 1 - 1 0

f(~)

1 dO

).

Therefore,

1/11~2~~;)~~; [m(p,f)+m(p, ~ ~2p1+2(p+1)!

-...::::

)]

• 2T(p,f)+0(1)

(p-r)'

p1+I·

It follows(by the assumption of p) that when p-oo, II converges to

Next fromlapl~p, Izl~r,

Up 1- 1 ~_1 1a z-p2 -I p2 zl-"'::::p-r' p

a.

and hence

Similarly, we have

By J(O)=t=-O,

J~n(t;O)dt

N(ep,O)=N(p,O)+ ~

n(p,O)

dt S T=n(p,O). ep p

Similarly we have N(ep,oo)~

Thus

n(p,oo).

°

uniformly.

1

25

THEORY OF NORMAL FAMILIES ~p!

p,+1 • N(ep,O)+N(ep,oo) p,+1 (p-r),+1

~p!

(ep),+1 • 2T(ep,f)+O(l) (p-r ),+1 (ep),+1

As before, one can derive as above that when p-oo, 12 tends to

°

uniformly. Conse-

quently, from (1. 1. 45), as p_oo, it is uniformly that rl,+1 rlz'+ 1 logf(z) =~l- 0'

. p!. (1. 1. 46)

The right side of (1. 1. 46) , as analysed above, is also convergent uniformly in any bounded domain containing neither zeros nor poles of fez). One can integrate both sides of the above equation from 0 to z, and notiJ1g that the right side can be integrated termwise. The p-th successive integration of( - O'p!

(Z_~)'+I

is

1 1 Z Z"I z-a +~+-;;r+ ••. +~ The p+ I-th integration becomes z-a

z

z,

Z2

wg(-)+-+-+···+-. -a a 2a 2 pa' Accordingly from (1. 1. 46), we have

Where P (z) denotes a suitable polynomial of degree no greater than p; it is produced through the successive p+ 1 integrations. Hence

fez) =e,(2)lim "...=

.

b

If Z=

°

. J'

z )er+"'+, Z Z' (1'-:'• 1h

is a zero or pole of fez) of multiplicity k, then by considering

by the above argument, we have

f~)

or ;!f(z) and

CHAPTER

26 z z z, (1--)e;;-+"'+, ap

{

P

1

}

141'

(1.1.47)

where the right side is convergent uniformly on any bounded domain containing neither zeros nor poles of fez). The numerator and denominator of the right side of to infinite products independently, so, for any z,

f

O. 1. 47) converge uniformly

(z) can be represented as a quotient

of two infinite products a necessary and sufficient condition is that both the numerator and denominator converge independently to finite values for all the values z. Now if lun

converge to finite values, then, clearly, the numerator and denominator of the right side of (1. 1. 47) converge uniformly to finite values in any bounded domain. By Pobson-Jensen formula, if lun ,,-co

Jr

"0

T(;;e t

dt is finite, then the above two infinite series COfl-

verge to finite val ues, this also proves the later part of the theorem.

2. The Normal Family of Meromorphic Functions 2. ,. THE CONCEPT OF NORMAL FAMILY Definition. Let F= (f.(z) }.El(I: index set) be a family of functions that are meromorphic in a domain D. F is said to be normal in D if every sequence {f. (z) } of functions belonging to that family has a subsequence that converges spherically uniformly in general sense inside D. Here we say converges spherically means that on the w-plane that touches tangentially at w=O of the Riemann sphere property that for any given e> 0,

L; of radius 1/2,

the values w of w=f(z) have the

there exists positive no (e, z) such that whenever m>

n>no, the corresponding spherical distance of two points fm(z) , f.(z) satisfying d (f.,(z) ,f.(z))

<e

{f. (z)} is said converges spherically uniformly meaning in general sense that if G is

an arbitrary closed subdomain in D, then the previous mentioned quantity no(e,z) will ofl-

27

THEORY OF NORMAL FAMILIES ly depend on E and e, but independent of the

Zl

s that belongs to G.

Definition. Let F be a family of functions that are meromorphic in D. F is said to be normal at a point

zE D if it is normal in some neighborhood of

z

0. e. ,in an open disk cen-

tered at z with sufficiently small radius). Theorem 1. 2. 1. Let F be a famuy of functwns that are me1'Omarphic m a dOllUJin D. A TlRCes-

8Ul'Y and suJJICIRnt conditwn far F to be naT'mal m D is that F Proof.

t,8

narllUJl at each point z m D.

The necessity is obvious.

We now prove the sufficiency. According to the hypothesis, to each point z in D corresponding a open disk C, such that F is normal in C,. We can construct a closed disk W, that lies inside C,. Then any sequence of functions {f. (z)} from F , a subsequence {f•• (z)} can be so chosen that it converges spherically uniformly in the general sense in W, I

noting that D can be covered by a countable subset {W.,> of the family {W.}(zED). Let {f.(z)} be an arbitrary sequence chosen from the family {f.(z)} , then there exists a subsequence {f1 .• (Z)} of {f.(z)} which converges spherically uniformly on W'I' Next from {f1 .• (Z)} a subsequence {f2 .• (Z)} can be chosen so that it converges spherically uniformly on W'2' And so on , in general, if a subsequence {f.-l .• (Z)} has been selected, then from it a subsequence of {f•.• (z)} can be selected so that the subsequence of (f •.• (z)} converges spherically uniformly on W'.' Obviously, {f•.• (z)} converges spheri-

cally uniformly on W'I

U W'2 U ... U W•• ' Now

choose, from the family of sequences {f•.•

(z) } (k= I ,2 ... · ,n= 1.2",,) constructed above, the sequence {f •.• (z)} (n=

1,2,,,,),

We are going to show that {f •.• (z)} converges spherically uniformly in the general sense on D. Let W be an arbitrary closed domain in D. Then a finite number of the W, s will COver D; let ko denote the largest subindex appeared on the W.s. Then clearly for

n~ko,

the corresponding subsequence {f.o" (z)} is a subsequence of the sequence {f•.• (z)}, therefore, it converges spherically uniformly on W.

1

UW, U·" UW, 2

10

and hence. on D.

2·2. THE PROPERTIES OF THE NORMAL FAMILIES OF MEROMORPHIC FUNCTIONS

Let d (WI' W2) denote the arc (interior one on the great circle) length of two points

28

CHAPTER

W' I ,W' 2 on point

WI

1

the Riemann sphere, which are stereographic projections of the two planar

,W2 respectively. Now let d(wl ,W2) denote the length of the chord WI' w'

2,

then it

is easy to verify that

Let

then d(WI ,W2) =-J(~1-~2)2+ (711-712)2+ «(;1-(;2)2 IWI- W21

-J 1 + 1WI 12 -J 1 + 1W2 12 1

I wl -

1 W2

1

O. 2.

1)

Define

and note

d(z, 0) =dO/z,oo). It follows that when (1. 2. 2)

or

(1.2.3) then

(1.2.4) or

0.2.5) We note that when

-

d (WI' W2)

< 1-

2

-Jl +A

, A> 1

(1. 2. 6)

2

holds then either (J. 2. 2) or (1.2.3) must be true. The reason is as follows: suppose

THEORY OF NORMAL FAMILIES

29

that neither 0.2.2) nor (1. 2. 3) is true, we may assume without loss of generality that

1

1WI 1~A, -1-1 ~A. Then W2

from

we have

1

= 1-

~ ---;:=:====

1+-1WI-1 12

1

1 A

1

2

This contradicts with (1. 2. 6). From the above observation we see that the statement: sequence {f. (z)} converges spherically uniformly in a domain D is equivalent to say that either If.,(z) - j.(z) 1 <e or

If)z) - j)z)

I<e must

be held for m>n>n(e,w), w is an arbitrary closed subset of

D, which also is equivalent to say that to any point

Zo

ED, there corresponds a neighbor-

hood _V(zo) so that either {f(z)} or {j)z)} converges uniformly in N(zo). It follows that if {f.(z)} converges spherically uniformly in a domain D, then its limit function is always meromorphic in D.

Theorem'.

2. 2.

If the sequence oj meromorphic junctions {f.(z)} converl}f!S spherically unir-

farmly to fo(z) on the closed aomain W,then the sequence

+

. afo(z) /l verges spherically umfarmly to vjo(z) +0'

PrOOf.

{~j:~:;t;}

(ao-/lv=J:.O) afro con-

CHAPTER

30

Now if

1ao-

1

py 1 #0, then to any w,

1+ Iwl 2

<[«00.

law+pI2+ lyw+ol 2

The theorem follows.

Theorem ,.

2. 3.

If the sequence {f. (z) } of meromarphlC functlO1!s converges spherically umform-

ly m the domam D m the general sense to functinn f 1!01!e

0,

then f is meromarphic m D. Moreover, if

of the functions m the sequence {f.(z)} assumRs the value a (jinzle or mfinzle) m D, then

the lunit Iunctwn f

0

also does not assume the value a, unless f 0 is identically equal to a m D.

Proof. The first half of the theorem has just been proved; the second half, according to Theorem 1. 2. 2, we only need to prove the case when a=O. We may assume that fo(z) ~O

and z be a zero of I. Then for a sufficiently small To, {f.(z)} converges uniformly

tofo(z) on Iz-zol~2TO. Furthermore, on Iz-zol~TO' fo(z) is regular and satisfies

1

1

1<2A for some constant A. On the circle Iz-zol=TO' there exists a constant 0, such that 1f 0 (z) I> 20. Now choose 1! sufficiently large, then according to the hypothesis of the uniformly convergent, it follows that 1f. (z) 10, and by the hypothesis of the theorem, f.(z) fo(z)

has no zero on

1 z-

Zo 1 ~ 2To; {f. (z)} converges uniformly to f 0 (z) on

1z

- Zo 1 ~ 2To.

Therefore, {fl .(z)} converges uniformly to fl o(z) on Iz-zo 1 =To. Now since on IzZo 1 =To, 1 f. (z)

1

>0,

and I' • (z) / f. (z) converges uniformly to

I'

0

(z) /10 (z), therefore

for sufficiently large nl, we have, for n>1Il .(z) _f' o(z) I<...!.. Iflf.(z) fo(Z) To

on 1 z- Zo 1 =

T.

It follows that ffl .(z) dz-~ffl o(z) dz 1<1 1_1 2m f.(z) 2m fo(z)

(1. 2. 7)

where the integration is along the positive direction of the circle. Viewing (1. 2. 7) in

< 1 z- Zo 1 <

which the first term in the absolute value is the number of the zeros of f.(z) in Iz-zo T,

which is zero, and the second term is the number of the zeros of

T,

which is

f 0 (z) in

1

> 1. Hence the absolute value of the difference of these two terms is always

THEORY OF NORMAL FAMILIES

> 1,

31

which leads to a contradiction.

Corollary 1. The ltmil functinn of a sequence of regular fundwns tlud converges spheru:ally unlr-

the general sense in a domain D must be a regular fundinn in D, unlRss u is identicaUy e-

form1y

UI

qual to

=.

Corollary

2.

Let {f. (z)} be a sequence of meromorphic functinns on a domain D. If {f. (z) }

cont'erges sph.erU:aUy umformly to a functIOn fo(z), which is not equal to constant a UlenticaUy and TIRl'eT'

takes the value a on the baundary of a closed suhdomain WCD. Then, for n>no, f.(z) and

fo(z) lim'e the same number of a-point in W.

Proof. It follows immediately from (1. 2. 7).

Corollary 3. Let F= {f(z)} be a family of fundinns that are regular m the domam D. If any

limit fUl1ctwII of F is not tdentu:ally equal to constant a, then, for an arbitrary closed subdmnam W CD, the

/lllll/berS

of the a-points of functinns in F are baunded a/)(Jl'f!.

Proof. If it is not bounded above, then a sequence of functions {J. (z)} CF can be cho-

sen such that f.(z) has at least n a-points in W. Now choose from {f.(z)} a subsequence

{f •.• (z)} which converges spherically uniformly in W, and, according to the hypothesis, its limit function cannot be a constant a. This contradicts with Corollary 2. Consequently, we have the following conclusion.

Corollary 4. Let F= {fez) } be a rumnal family of merollwrphic fundwns in dmoom D and let E

be a closed subdmnam of D. Let Zo be a pomt in

w.

If the value a is not a cluster point of the set

(f(zo)}, then, the numbers of the a-points of the fundinns in F that belong to are baunded abave.

Theorem 1.2.4. Let F= {f.(z)} be a rumnal family of mermoorpltic fundinns in domain D.

If a sequence (f. (z)} CF converges sphericaUy uniformly in a subset W of D which has a limit POint tn D, then {f.(z)} converges sphericaUy unifarmly in the general sense m D. PrOOf. Since F is normal in D, it follows that a subsequence {f •• (z)} of {f.(z)} can be Chosen that converges spherically uniformly in D in the general sense. Let fo (z) be the

CHAPTER

32

1

limit function of {J., (z) }, which, according to Theorem 1. 2. 3 is merom orphic in D. We now show it converges spherically uniformly in the general sense to fo(z) in D. Suppose it were not so. Then it would be the case for some closed subdomain WCD. It follows that there exist an infinite sequence of points {z.} CW, a subsequence {f.: (z)} of {f. (z) }, and a positive number e such that d(J:, (z.) ,fo(z.) ):;::e.

(1. 2. 8)

Now on the other hand a subsequence {f •• (z)} of {f:, (z)} can be selected that will converge spherically uniformly in W. Let fO (z) be the limit function of {f., (z) }, then by the hypothesis fO (z) and fo(z) coincide in W. Since fo(z) and fO (z) are meromorphic functions, it follows that fo(z)=fO (z) in D. This contradicts with (1. 2. 8).

Definition (Spherical Equicontinuity). Let F= {f.(z)} be a family of functions that are meromorphic in a closed domain W. If, to each given e>O, there exists a 6(e»0 such that for any two points Zl ,Z2 in W satisfying iZ,-Z2i <6(e), the following inequality d(f(Z,),f(z2))<e

holds for any function f in F, then F is called spherically equicontinuous in W.

Theorem

'.2. 5. (Ascoli).

Let F= {f.(z)} be a sequence of meromnrphic funct!al/.8 defuwd

UI

a domam D that i.s spherzeally equzrontinuous in a cwsed subset W of D. ThRn Ii i.s poss!ble to select from F a subsequence that converges spherzcaUy uniformly in W.

Proof. Choose a sequence of points Zl ,Z2, "', in W, which is also dense in W. Since for

each point Zj E W, the sequence {f. (Zj)} has at least one cluster point on the Riemann sphere

2:,

therefore, from it a spherically convergent subsequence {fl .• (z)} can be cho-

sen. Generally, if a sequence of meromorphic functions {f.-, .• (z)} has been defined, then a subsequence {f •.• (z)} can be selected, so that {f •.• (z.)} is spherically convergent. Now consider the sequence {f •.• (z) }, it converges spherically at each point of the 1>et {Zj}. We are going to show that actually the sequence {f •.• (z)} spherically uniformlY

converges on the entire set W. Let e>O be given. Let C.denote the open disk of radius 6(=e/3) with center at z" Then a finite many of such disks will cover W, let them be denoted C., ' C.2 , , · , C.,. Note that {f ••• (z.)} is convergent for each z.;' Hence when m>n> N.;. we have

THEORY OF NORMAL FAMILIES

33

0.2.9) Choose

~V=1TI£lxN •• l~i~,

Now to any

zEE, it belongs to some

C •• Hence

I

,

It follows, from the hypothesis of spherical equicontinuity,

d(f •.• (z) ,

1

f •.• (z.,»<3e,

(1. 2.10)

and

0.2.11) Combining(1. 2. 9), (1. 2. 10), and (1. 2. 11). we obtain, for m>n>N, d(f •.• (z)

,f..... (z) )<e,

where .:Y depends on e and W, but is independent of z. This shows that {f •.• (z)} is spherically uniformly convergent on W.

Theorem ,. be normal

~n

2. 6. A necessary and sufficiRnt cmulitUlTl far a family of mermnarphic functinns F

to

a domam D is that F is sphl!rically equicontin1U.JUS on arbitrary cwsed suMmnam of D.

Proof. The sufficiency follows from Theorem 1. 2. 5. We only need to prove the necessity. Suppose that F is not spherically equicontinuous on a certain closed subdomain D. This assumption implies that there exist a positive number e

> 0,

we

two sequences of

points {z,} and {z'.} in E, and a sequence of functions {f.(z)} in F such that

Iz.-z' .I<-.l, n but d(f.(z.) , f.(z' .) )~e.

Since {z.}

ew,

(1. 2. 12)

it follows that a subsequence {z•.l of {z.} will converge to a point Zo E

W, consequently the corresponding sequence {z' • } satisfies z, • -zoo On the other hand,

•

•

since F is normal, the sequence {f.,} contains a subsequence {f:,} which converges spherically uniformly to fo(z) on W. Now we rewrite {n.' } to be {n}, hence, we may assUme, at the beginning, that and

CHAPTER

34

1

f.-fo

uniformly, hence, when n>no(e/3) d(f.(z.) ,fo(z.))<

T1 e,

d(f.(z' .) ,fo(z' .))<

T1 e•

0.2.13) (]. 2.14)

From (1. 2.12), (1. 2. 13), and 0.2.14), we obtain

1

d(fo(z.) ,fo(z.' ) »Te•

But fo(z) is meromorphic on the Riemann sphere, when Z.---'Zo,

z.' ---..zo

we should have d(fo(z.) , fo(z' .) )-0.

This is a contradiction. Hence F must be spherically equicontinuous on closed

subdomain~

of W.

Definition. The quantity lim

d(f(zo+ ~z) ,f(zo))

I~zl

,:,.-0

is called the spherical derivative of fez) at z, and is denoted by Df(z). When f(zo) #00, it is easy to see that

__ 1f'(zo)1 DfC"o)-I+ If(zo)

When f (zo)

= 00,

12 •

(1. 2.15)

we define Df(zo)

= lim

Dj(z).

Let L. (z) represents a linear function corresponding to a rotation of the Riemann sphere. Then

it follows that

THEOR Y OF NORMAL FAMILIES

35

Theorem 1.2.7. If fez) is me'T'mnorphic on a clnsed amoom W, thRn Df(z) is contmuaus and baUruled an W.

Proof. Let us first divide E into a finite number of subsets W;: WI> W 2' ••• , W., such that no two distinct zeros or poles of f will fall into the same subset. However, in each W;, either f (z) or f (1z ) is regular, and hence is continuous and bounded there. Now from the observation Df(z) =D(l/f(z)), one concludes easily that Df(z) is continuous and bounded on W. Theorem 1. 2. 8. Let {J. (z)} be a sequence of mermnorphi.c functions that are aefWRd on a closed aomam Wand converges spheri.cal1y unifarmly to a rneromarphic function fo(z) on W. ThRn

{Df.(z)} converges unifarmly to Dfo(z) on W. Proof. As in the previous proof, we divide W into W; s of finite closed subdomains such that fez) or f(1z ) is regular on each of the W;s. If fez) is regular on W, then, for sufficiently large n, f.(z) is regular on W. Thus {J' .(z)} converges uniformly to f' o(z), If'.(z)1 . If'o(z)1 1. and hence, 1+ If.(z) 12 converges umformly to 1+ Ifo(z) 12' If fo(z) IS regular, then

1 . 1 1 f.(z) converges umformly to fo(z)' It follows that D f.(z) converges to Dfo(z) and, hence, DJ.(z) converges uniformly to Dfo(z). As this holds true on each of the W; s, therefore, Df.(z) converges uniformly to Dfo(z) on entire W. Theorem 1.2. 9.(Marty). Let F= {J.(z) }.El(J: index set) be a family of functions that are merO/norplac m a a071lnm D. ThRn a necessary and sufficient conduwn for F to be norrTUJi m D

/,8

that to any closed sulxlO/oom W of D, there exists a constant M (W) (aependmg an W only) such

that to any f E F the following condition holds: Df(z)<M(W) , zE W.

Proof. The nesessity. If the stated condition does not hold, then, a sequence of functions (f.(z)} and points {z,} can be chosen from F and W respectively such that Df.(z,»n.

(1. 2.16)

Now choose from {J. (z)} a subsequence {J•• (z) }, which converges spherically uniform-

CHAPTER

36

1

ly to fo(z) on W. Since fo(z) is merom orphic on W, it follows that Dfo(z)<MoJ zEW,

for some constant Mo. On the other hand, because Df.(z) converges uniformly to Dfo(z) , we have, when n>no with no sufficiently large,

+ 1.

Df.(z)<Mo

This contradicts with (1. 2.16). The sufficiency. From the definition, we have d(f(zl)

,f(z2»~

f'2 Df(z) Idzl< IZ2-ZII • M(Do) , "I

(the integration path is along the segffient [Zl' Z2 J). This shows that (fa(z)} is spherically equicontinuous on W. Since W is an arbitrary closed subdomain of D, it follows from Theorem 1. 2. 5 that (fa(z)} is normal in D.

Theorem ,. bounded

In

2. , o.

Let F

=

{fa (z)} be a family of functiDns that are requ1ar and uniformLy

a dOIT/nin D. Tlu!n F is normal in D.

Proof. Let us choose an arbitrary point Zo from D and draw a disk: Iz- Zo I~ro completely

contained in D. Then, to any fEF, we have If(z) Zo I ~no, I

~

If ("')

It follows that whenever

1

__

r

1-12n\_zoi~2TO

I<M,

for zED. Hence when

Iz-

f(t;) M (t;-Z)2 dt; I~ro

Iz- Zo I ~ro, M

Df(z)~-.

ro

By virtue of Theorem 1. 2. 8, (fa(Z)} is normal on

Izo-z I~ro.

Hence it follows from

Theorem 1. 1. 1 that (fa(Z)} is normal in D. We note from the proof the above theorem that the conclusion of Theorem 1. 2. 10 remains valid if the condition" (fa(Z)} is uniformly bounded in D" is replaced by" {f. (z)} is uniformly bounded in arbitrary closed subdomains of D".

Corollary 4. Let F= (fa(z)} be a family of functwns toot are regular in a dOIT/nin D. If F uniformly bounded in closed suhdomains of D, then F is normal in D.

lS

37

THEORY OF NORMAL FAMILIES

corollary 5. Let F= {fo(z)} be a family of functinns that are 7T/R'1'ClTTUJT'phic in a domain D. If

to each fEF and zED, If(z)-a l~m>O, for

SOlllR

value a; namely F fails

to COV(!1' a C(!1'tain regwn(non-empty) on the Rwnann sphere, then

F is normal in D.

Proof. This follows immediately from Theorems 1. 2. 9 and 1. 2. 1. Corollary 6. (Vitali). ut F= {f.(z)} be a sequence of fUnctinns that are 71I(!1'omorphic in a d~

lTIain D. Suppose that there exists a vabJe b such that (z)} conV(!1'ges on a

{f.(Z~-b}

!8

uniformly bounded. If {f.

set of points of D that has a cWster point in D, then {f.(z)} converges spher/r-

cally umformly in any closed subdomain of D (or equivalenily to say that it converges in the

i,nt(!1'inr

of D).

Proof. This follows as a consequence of Theorems 1. 2. 4 a'ld 1. 2. 10.

2. ". ut F= {fo(z)} be a family of functinns that are regulur in a domain D. If at any arbitrary point Zo ED, {f (zo) } is bounded, say by a constant, then {f (z) } is uniformly

Theorem ,.

0

0

bounded on closed subdomains of D.

Proof. Suppose that F is not uniformly bounded on some closed subdomain W, this means that there exist a sequence {f.(z)} of functions f E F and points {z.} CW such that

If.(z.) I>n.

0.2.17)

But {f.(z)} contains a subsequence {f., (z)} that converges spherically uniformly to a function fo(z) on W. Since If.,(zo) I<M, it follows from Theorem 1. 2. 2 that fo(z) is regular on W, and, hence, for all z

E w,

Ifo(z)

I
for some constant K. Conse-

quently, for sufficiently large ko, whenever k>k o

If., (z)

1< Ifo(z) 1+ 1
However, this will be incompatible with 0.2.7) by choosing n>K+1. The theorem is thus proved.

CHAPTER

38

1

3. The Distance of a Family of Functions at a Point Let F= {f.(z)} be a family of functions that are meromorphic in a domain D. Consider sup{d(f(z)) ,d(f(z2)): f E F, z) ,Z2 E {z: 1Z-Zo 1
0,( {f} ,zo,a).

Note that 0.3.1) decreases montonically as

adoes.

Hence when

a-O,

the limit of

O. 3. 1) exists, and the quantity lim 0,( {f} ,Zo ,a} =0,( {f} ,zo)

0.3.2)

6-0

is called the distance of F at point z. Theorem ,. 3. ,. A 1U!Cessary and sufficiRnt condition far a famuy of 1TU!T'omarphic fUnctions F defmed m a dorooin D that are spherically equicantinuaus on subdorooins of D ts that the dtstance of F at an arbitrary poUlt of D is zero.

Proof. The necessity is obvious. We shall only prove the sufficiency. Let Zo be an arbitrary point in D, then 0, ( {f } , zo)

=

O. Now if F is not spherically equicontinuous on

some subdomain W of D, it implies that there exists e> 0, and sequence of functions {f. (z)} and sequence of points {z.} can be chosen from F and W respectively such that

d(f.(z.) ,f.(z.' ) )~e

(Iz.-z.'

1 1<-). n

o. 3. 3)

A subsequence {z.,} can be chosen from {z.} that converges to a point Zo E W. This will z~. -zoo

yield

Note the fact 0, ( {f} ,zo) = 0; we have, for arbitrarily small a> 0, to any

fEF whenever

Iz'

-zol
Iz"-zol
this contradicts with (1. 3. 3).

Corollary. Let F= {f.(z) } be a family of functlO1!S that are me7'OmarphIC ill a domain D. A

1/RC-

essary and sufficiRnt condition far F to be narmal in D is that the distance of F at each paint of D ts zero.

Thus if a family of functions F= {f.(z)} that are meromorphic in a domain D is not normal in D, it implies from Theorem 1. 2. 1 that {f.(z)} fails to be normal at some point Zo E D. This also means 0, ( {f} ,zo) point.

> 0,

and such a point Zo is called an abnormal

THEORY OF NORMAL FAMILIES

39

Furthermore, if the family F= (f.(z)} is not normal at a point zo, then, according to the definition of normality, in any open disk C6 :

{

IZ-Zo I <6}

there exists a sequence

of functions 8 6 = (f.(z)} from F such that none subsequence of (f.(z)} can be convergent spherically uniformly in C6 • However, generally speaking, the subsequence 8 6 mentioned above is dependent on

o.

But if 8. is independent on 6, that means, in any open disk

Iz- Zo I<6'

, any subse-

quence of S6 will not be convergent uniformly, and such a point Zo is called an 0- point;

an abnormal point, is generally called a J -point. In the investigation of the normality of a family of functions in general, a J-point may not be an O-point; however, for family of meromorphic functions each J-point is also an O-point as can be shown by the theorem below.

Theorem

'.3.2.

Let F= (f.(z)} be a family of fUnctions that are lnerolnarphic

D. If F falls to be normal at a point zoE D, then there must be somR

SRf[Ul'TtCR

In

a domain

of fWitians {f.

(z)} of F such that none subsequence of (f.(z)} will be convergent SpherUXlUy umformly in

<e for

Zo I

Iz-

arburary 6>0.

Proof. Since O.({f},zo)=w>O, a sequence of functions (f.(z)} and sequences of

points {z.} and

{z~}

can be chosen such that

Iz. -

Zo

I< -.L, Iz~ n

Zo

I< -.L, n

and

1

d(f.(z.) , f.(z'.))>Zw.

The fact that meromorphic functions are continuous spherically and the condition Z.-Zo, z' '-=0 indicates that none of the functions in (f.(z)} can occur infinitely many times. It follows that any subsequence of (f(z)} will not be convergent spherically uniformly in

z- Zo I ~o; 6 an arbitrary positive number. We now know that when (f.(z)} is not normal at point zo, then (f.(z)} will not be spherically uniformly convergent in a neighborhood of zoo However, we note that there may be a subsequence of (f.(z)} convergent spherically uniformly in a neighborhood of zoo For instance,

ty on

Iz I =

a function f (z) that is regular in Iz I< 1 having at least one singulari-

1 can be represented by a Taylor series:

CHAPTER

40

1

fCz)= ao+alz+···+a.z·+···,

which converges uniformly in the general sense 0. e. , in closed subsets of Iz 1< 1). It fol-

.

lows the sequence {S.Cz)}

= { L>.;!} .=0

of the partial sums of f converges uniformly in the general sense in Iz I

< 1.

But it is pos-

sible that there is some subsequence {S., Cz) } of {S. Cz) } satisfying suitable condition will be convergent at a neighborhood of a point on the circumference

I z I = 1,

say z = 1.

When this phenomenon occurs it is called overconvergent.

Ostrowski's theorem. Let

~a'z'

fCz)=

• -

be regular in

no
Iz I< 1,

0

and, mureover, it is regular at a point Zo on

Iz I =

1. Let e> 0 be

gwen

and

< ... O+e)nU-1

Suppose now that whenever n salisfiRs

n2t-1

Ck=I,2, .. ·).

< n< n2' Ck = 1 , 2 , ••• ) then a. = 0 in the above expan-

sinn. Then '21+1

{S'Cz) } = { ~ a.z' } • = 0

converges unifarmly in a sufficienay small neighborhood of zoo

Pr()()f. Choose integer

p> l/e,

and let 1

z=2zoC~'+~'+I) ,

fCz)

then determine bo , bl

, ... ,

=gC~),

b.. , ... , so that the following equations hold:

Ck=O,I,2· .. ). Thus 00

"21+2- 1

gC~)= ~ ~ .t --

0 --"21

b.. ~

THEORY OF NORMAL FAMILIES

Noting that

~0 Ia.1 z·

Iz I < 1,

converges in

• =

Iz I< 1. It follows

is convergent in

41

we can conclude that

that (1~1<1)·

On the other hand, we can show that g(~) is regular on Izl~1. The reason is that under

I~ I ~ 1 ,

only ~= 1 can lead to

zl<1; namely g(z)=

I~'+~'+I I = 2,

bb.. ~ is regular on

,"=0

1~Il).

Izl~1,

and hence

Z=Zo.

If ~¥-1, then

I

therefore it converges uniformly in

Hence co

g(~)= b

"21:+2- 1

b

b.. ~,

.t = 0 "="21:

or {S.,. (z) } converges uniformly in a region corresponding to hood of

I~ I<1' which is a neighbor-

Zoo

From the theorem one can immediately derive the following result. Gap therorem (Hadamard). Let

f(z)=ba.z" " = 0

be a power sertes havmg

lim

A.+ 1 >

~A.

1, then

Izl<1

Iz I = 1 is

II

as lis circle of wnt'e7'gence.lf O~AI
the natural boundary of

f.

Proof. By Ostrowski's theorem noting (n21=n2t+1 =1,1), if f is regular at a certain point I

on

1

z 1 = 1, then {S., (z) } =

{b

a.,z·, } will be convergent uniformly in a neighborhood

• = I

of that point. However the convergence of {SA (z)} is nothing but in the sense of the con-

•

vergence of the Taylor series, which is impossible to be convergent at anywhere in

Izl>1.

Thus fez) cannot be regular at any point on

Izl =1.

Theorem 1. 3. 3. Let (f.(z)} be a family of functinns that are 7neTQ/1UJT'phic in a certain domain. Then at any of lis ab7wrrMl point zo, 0.( {f} ,zo) =w=

;.

CHAPTER

42 ~VofR: Thus

1

a family of merom orphic functions at any of its abnormal point has not

only w>O but also is impossible to have O<w<; ,i. e. w has to be equal to n/2.

Proof. Assume that w 0 such that w+ 2TJ
E

sis, we have, for any f

{f.(z)} and sufficiently small 0,

d(f(zo) ,fez) )<w+TJ (Iz-zo I<0).

Choose {f.(z)} a subsequence of {f.(z)} such that {f. (zo) } converges spherically to

WOo

Then for n>no(TJ) d(f.(zo) ,wo)
Hence under Iz-zolno(TJ) , we have d(wo,f.(z»<w+2TJ; namely {f.(z)i does not assume any value w that satisfies d(w,wo»w+2TJ. It follows from Theorem 1. 2.10 and its corollary 2 that {f.(z)} and, hence, {f.(z)} is normal at z,which is a contradiction. Thus we must conclude that w=n/2.

4. On Meromorphic Functions with Deficient Values Theorem ,. 4. , .(Schottky). If f (z)

18

reqular and does 1Wl assume thR values 0 and 1

~n

Izl
0.4.

1)

Moreov(!J' ,whRn If(O)

1>0, If(O)-II>o,

1 If(O) 1<0,

Q(f(O) ,O)~Q] (0,0).

Where

Q, Q]

(1. 4.2)

are constants that depend on thR quantities appeared in the brackets correspondmg only.

Proof. Let w]=0,w2=I,w3==' then n(r,w;)=N(r,w;)-O

(i=I,2,3),

T(r,j)=m(r,j).

By Theorem 1. 1. 13 we have m(r,j)
(1. 4. 3)

(r
We first set oo=min( If(O)

I

I , !f(0) -11, i/O)

I)

(oo~l)

,

THEORY OF NORMAL FAMILIES and consider

II'

43

(0) I::;t: 0 and note C_~=/(O), G(z)=/(z)(f(z)-l),

Go=/(O)(/(O)-l), d=O, mo=O(Go::;t:O,::;t:oo) , P_~=I' (0).

By applying Theorem 1. 1. 13 and from 0.1. 40), (1. 1. 24) we have, for R>p>r>

ro, with suitably fixed ro> 0, m(r,f)
1 + 2Log+-p-r

+3Log+ R+ 4log+-.L+ 6 (log + log + R+Log+ Log+-.L) ro ro

;0 + 16log+log ;0 + 10Log3+20Log2+ 2kJ +Log II' ~O) I<6Log+m(p,f) + 2log + p~r + llLog

+[{J (Uo,R,ro)+log,

I'

~O) ,.

After some calculations we can obtain the following relation:

R>r>mllx(ro ,R-1),

1

m(r,f)~Mo+2{2+Log2+2Log R-r

+ [{

J

(00, R, ro)

+ log , I' ~ 0) , }.

(1.4.4)

Note in which M is a constant that satisfies

1

x-12log+x~2x, x~M.

It follows that if

II'

(0)

I~ :~,

then

m(r,f)~Mo+2{2+Log2+2Log --R 1 +[{J(oo,R,ro) + r However, if P.

II'

II' (z)

(0)

I ~ :~,

I <:~

in

log :R}. uo

then two cases will be considered separately:

Iz I
this case, we have, for

I/(z)I=I/(O)+

. JI'

Iz I
(z)dzl

o

~ 1/(0) 1+ hence the theorem follows clearly.

1 2 S II' (z) Ildzl
0

(1.4.5)

CHAPTER

44 2°. If' (z)

I
in Izl
I =g~

for some point a on

1

Izl =

A. In this case we choose r satisfying R>r > A and set

g(Z)=f(r2(Z-~») az-r

,

0.4.6)

g(O)=f(a) then Ig(O)-f(O)

1= If(a)-f(O) I = I

J>'

(z)dzl

~ Ia I g~ < ~ 00' Consequently 1

Ig(O) I~Zoo,

1

Ig(O)-ll~Zoo,

I9 ( 0 ) I~ ;0 + ~ ~ ;0 ' 2A2 x 2A2 19'(O)I=lf'(a)( r2 =;~r r2 ' moreover, 9 (z) is regular and does not assume 0,1 in

O. 4.

Iz I< r~C::r~) .

This becomes exactly the same case as (1. 4. 4) (by setting Rl =

r~C::r~) ),

R1>r>rruzx(R1-l,ro)

1

m(r,g)~Mo+2{2+log2+2log R1-r

+f{l(~ ,R1,ro)+109Ig' ~O) I}· Now, since R>RH if n>max(R-l ,ro), then RA+r 2 m(r,g)~Mo+ {2+log2+2log r(r-A) (R-r) 00 2r2R } +A1(z,R1,ro)+log 00(r2-A2) . T

But from 0.4.7), we have r+A

m(r,J)~-Am(r,g)

r-

hence, for

r>max(R~A ,R-l ,ro), 4R

7)

,

we obtain

3R

m(r,f)~R_r {Mo+2(2+log2+2log (R-r)

Hence, for

45

THEORY OF NORMAL FAMILIES

+[{)(~ ,R),ro)+log oo(:~r)J}.

(1. 4. 8)

Noting that m(r,f) is an increasing function of r, we have for

R>r>max(A,R-l ,ro), R+r

12R

8R

m(r,f)~m(-2-,f)~R_r {M o+2(2+log2+2Iog (R-r)2

• 00 2R2} +A)(2,R,r o)+log oo(R-r)J •

Thus when Izl~A,

Hence, for r>max(R-l ,ro), we have

1 00 m(r,f)
8R + R-r {Mo

12R • 00 2R2} +2(2+log2+2log (R_r)2+ A )(2,R),ro)+log oo(R-r)J ,

(1.4.9)

and, moreover,

loglf(z) Therefore if we set r=

r+ Izl l
R+lzi 2 ,then from

(r>

Izl).

(1. 4.5) and 0.4.9), and for

Izl
we have

0.4.10) Here we note that

[{2

is bounded when 1/(R-r) does. The theorem follows immediately

and consider f(Rz) instead of fez) and by observing that

[{3

is independent of R.

Corollary'. The family F= {f.(z)} of functinns that are requlm and do not assume the vahtes 0,1 in a domain D is nurmal in D.

Proof. By virtue of Theorem 1. 2. 1, it suffices to show that F is normal at each point of D. Now choose an arbitrary point Zo and sufficiently small ro such that Iz-zo I
(z)} of {f.(z)} such that in Iz-zol
CHAPTER

46

If., (zo) I>00, If., (zo)-II>oo,

I

f.,

1

~zo) 1>00•

Since f.,(z) is regular and#O,I, we have, for Iz-zol
Accordingly, we can choose from {f., (z) } a subsequence which converges uniformly in Z-Zo I<,0' 1'0(0<0' <0), this shows that {fa(z)} is normal at zoo

2. Suppose that the sequence {f. (zo)} has no cluster points other than 0, 1. This means that there exists a subsequence {f., (zo)} converges to one of the values 0,1 and oo. We may assume without loss of generality that 1 is the value, the other cases can be

obtained by simply considering 1- fez) or 1-1/ f(z), we recall here that linear transformations preserve spherically uniformly convergent property, and hence, the normality. Let g.,(z) =

logf. (z)

+ 2ni

'4 n,

logf., (z) =uo+ivo

' (-n
then when k--oo, g., (zo)-- 21 . But g., (z) # 0,1 , oo, as argued in the above, there exists a subsequence {g., (z)} which converges spherically uniformly to go (z) in

Iz- Zo I<.0' To

(0<0' <0), we are going to show that go(z) has to be equal to 1/2 identically. By the

hypothesis that f., # 1, it follows that logf., (z) # 0, and hence g., (z) does not assume

1/

2. It follows by Theorem 1. 2. 3 that go(z) = 1/2 identically. Now since {g., (z)} converges to a constant 1/2, therefore

converges uniformly to constant 1. The theorem is thus proved. Corollary

2. A

family of functions F= {fa(z)} that are merOlnarphic in a domain Dis narmal tn

D if there are three d!.Slinct values OIl/its

WI , W2

and

W3

(may include oo) such that any functwn f m F

the three values m D.

Proof. This follows immediately from Corollary 1 of Theorems 1. 4. ] and 1. 2. 2.

Note that from Corollary 2 above we deduce easily that if a family of functions F= {f. (z)} meromorphic in a domain D is abnormal at a point zoE D, then each function in the

family F assumes every vaue, except possibly two exceptional values, infinitely many times in an arbitrary neighborhood of Zo in D: Iz-zo I<0(0)0).

THEOR Y OF NORMAL FAMILIES

47

Note also that Picard theorem follows immediately from Theorem 1. 4. 1. Suppose that fez) is an entire function and omits two finite values, say 0 and 1, then to any given R, we have If(z) I
( Izl
Since R can be arbitrarily large, by Liouville's theorem fez) must be a constant. We have emphasized to unify the methods in our presentations; for instance, we derive Theorem 1. 4. 1 from Theorem 1. 1. 13. In generally speaking Theorem 1. 4. 1 (Schottky theorem), and, hence, Theorem 1. 1. 13 can be proved most precisely by utilizing modular functions. It had been proved earlier by Landau, Bloch, and Valiron by using some elementary methods. Among them, the most simplified one is the so called Bloch-Valiron-Landau method. Although it bears no important connection to the present book, however, the proof has its own merits and has something to do with the conformal mapping, we would like to illustrate the proof as follows.

Theorem 1. 4. 2.(Bloch).I.etfunctwnw=f(z) berequJman Izl~l wahf' (O)=I.Thenf (z) assumes every vahu! of an open disk of radius B;here B

t8

Proof. We may assume without any loss of generality that F(O)=O· max If' (z)

a constant uuJependent of fez).

f (0) = O.

Set

I, Izl~l-O.

Then F(O) is continuous on the interval O~O~l with F(O)=O,F(l)=1. It follows that there exists some 00 with O
Since

ITUlX

1*;;11-°1

If' (z)

I is attained

on

Iz I = 1 -

0, there are some

Zo

with

IZo I = 1 -

Such that

1

If' (zo)I=Oo·

Now from the fact that

Iz- zo I~ ~

is contained in

. rna:. If'

0 0 ="2 00 F("2)

Iz I~ 1- ~ I

(z) <1,

I.I<:I-y

We have 2 If' (z) 1<00

;

Iz-zol~~ •

entirely and

00

CHAPTER

48

1

1 Hence, for Iz-zol~200, If(z) - f(zo)

I= I

Now we proceed to show that, in the disk:

J:. f' (~)d~ I<1.

Iz-zo I<~

fez) assumes every value in

the disk IW-f(zo) 1<116 •

m fact,

if f(z)=j::.wo

in

Iz-zol<~,

then

f' (z) 2( wo- f( Zo ))(z-zo)+···

h(z)=

is regular in

Iz- Zo I< ~ , moreover, If(z)-f(zo)

2

I

1

Ih(z)I~I+ Iwo- f( Zo )1 ~1+1 wo- f( Zo )1· It follows that If' (Zo) 12 1+ 4I w o-f(Zo)

12

• O~ -.l J2"1 2 I __ 00 .'1' 4 ~2:1T 0 h (~) d
~1+1 wo- f ()I· Zo This yields IWo - f (zo)

I~ 116. Thus we have not just proved

clude the absolute constant B, at least,

~1/16.

As a matter of fact, the disk of radius B in Theorem a certain subdomain in

Izl <1

the theorem but also can con-

under the mapping fez)

I

1. 4. 2

is a univalent image of

it indicates particularly that z=


prove the Schottky theorem, it suffices to show that fez) assumes every value of a certain disk of radius B in the w-plane. By using some theorems about univalent functions in the later sections (Theorems 1-

6.3 and

1. 6. 5),

one can obtain Theorem 1. 4. 2 more precisely as follows.

As before, choose Zo with

IZo 1= 1-00 such

If(z)-f(zo)

Set

1<1,

that f' (zo)

=

Iz-zol~~.

;0 ' Ie 1= 11

49

THEORY OF NORMAL FAMILIES

then g(';) is regular in 1~1<1 and satisfies Ig(~) 1<2, g(O)=O,g' (0)=1. According to Theorem 1. 6. 5, g(~) is a univalent function in 1~I
h(~) =J.-g(P2~)' P2

then h (0) = 0, h' (0) = 1. Moreover, h(~) is a univalent function in 1~ 1< 1, it follows (from Theorem 1. 6. 3 by letting P2 (1 ) =

+)

that the image of 1~ 1< 1 under the map-

ping h (s) contains an open disk of radius P2 (1 ). Therefore, the image of 1~ 1< 1, under 9 (t;) contains a univalent disk of radius fP2 (1 ). It follows that the image of the disk 1zZo

1< ~ under the mapping fez) contains a univalent disk of radius P2 fP;( 1). The least upper bound of the B' s under the above more precise form is called the

Bloch constant. We now apply Bloch theorem to prove Schottky theorem as follows. It suffices to show the case that R= 1. If fez) is regular and omits 0,1 in 1z 1<1, then

g(z) =kJg{

J

kJgt;;) -

kJgf(~) - I } 2m

(1. 4.11)

is regular and omits the following set of values in Izl
r:--1

nni

±u.... =±kJg(v m +vm-1)+2' m=I,2,""

(1. 4. 12)

n=0,±I,±2,.. •.

Here, when z= 0, imaginary part v of the logarithm is chosen to satisfy -n
J.

Since fez) omits 0,1 in Izl
1<1. Hence, both functions

Wg2f

1. Clearly, this function does not assume the value zero there, and hence, 9 (z) is regular and omits the set of values (1.4.12) in Izl<1. Now by solving (1. 4.11), we obtain (1. 4. 13)

If ~ r:--1 nni g(z)=±kJg(v m +vm-1)+2'

then

CHAPTER

50

fez)

= -e±f(..J::+L)2+<..J::-£h = _e±wi(2.. -1l = 1.

On the other hand the set Z =

{± u.... }

distributes on the entire u>-plane I the distances

.. =1

of two adjacent points of Z on the real axis are U .. +I.O-U...

o=iog

-v'm+l+-hI ~ ~ .....; m +.....;m-1

These set of values attains its maximum log ( f t

+ 1) at m = 1.

The set Z can be ob-

tained by adding the set {±;1I'i} ~ 1 to each point of Z that lies on the real axis. It follows that centered at an arbitrary point in the u>-plane draws a circle of radius larger than log ( f t

~

I

+ 1 ) + ~i I (particularly a circle of radius 1) that will contain some points from

the set Z. We calculate g' (z) the derivative of g(z). If for some 0 such that g' (~)#O in I~

I<0,

then the function h(~) =g(~+ (1-0)z) .. (1-0)g' (~)

is regular with h' (0)=1 and omits all the values U....

E E. Thus any circle with a radius

(1-0) ~ g'

(1-~I;;'(~)1

in

Izl
where

(~) I in the u>-plane will contain at least

one such omitted value. It follows that

(1-0)~g' (~)I>B~/6'

Ig'

(~)1
Consequently, in Izl
Ig(z) 1< Ig(O) 1+ / that shows

I9 (z) I

600 ==K4 (f(O) ,0),

is bounded above by a quantity depending on 9 (0) (hence on f (0)

and 0). Hence by (1.4.13) for Izl
THEORY OF NORMAL FAMILIES

51

This also is the so called Schottky theorem. Note that from 0.4.13) we see that if f(O) does not near 0 or does not near

00.

Hence if

6< If(O) 1<

!'

00,

then Ig(O)

I

then Ks(f(O) ,(})
in the meantime 1- fez) will behave similarly to that of fez) and will not assume values

I
0,1,00. We obtain If(O)

and hence Ks(f(O),(})
most general form of Schottky theorem. Theorem 1.4. 3.(Schottky). If fez) is regular with If(O) I<M and omits vahles 0,1 in 1<1, tlien If(z)

1
Izl<(}R, here

!J2 is a constant deperul.mg

only on M and

O.

Theorem 1. 4. 4.( Landa u). Let f (z) be requ1ar with fl (0) ¥ 0 and omits values 0, 1 m Iz R. Then R

IS 1W

Proof. Let 0=

greater than a quantity that deperul.s only an f (0) and I fl (0)

~

, then by virtue of Theorem 1. 4.3, we have, for If(z) I
~

Iz

I<

I.

Iz I
)=K(f(O)).

Also from fl

f

(O)=~

2m

R

1"1- 2

f(Pd~, ~

we derive

If

I

(0) 1~2K(f(0))

-.::,

R

R~2I{(f(0)) ,-.::, Ifl (0) I

Corollary. If fez) is regular and omits 0,1 in Iz I
depends only on f (0),

po) (0),

po) (0) ¥

0, then R is bounded by a

and n.

Proof. The proof will be exactly the same as in the proof of Theorem 1. 4. 4. From

PO)(O)=~

2m

it fOllows that

f

f(~)d~,

1"1-';' ~+1

CHAPTER

52

n! ([{(f(0)) f(·l (0)

There are several generalizations of Theorems 1. 4. 3 and 1. 4. 4. We state some of the main results as follows. Let fez) =ao+alz+ ••• +a.z"+ •••

Iai I ~[{i (i= 0.1 .2 ••••• n) and assume the respectively. in Izl
be regular with (n~m) time.

If(z) 1~.Q.... ([(o'[(I'··· .[{•• O.R).

1. Here

values O. 1 no more than n, In

Q .... ([{ O. [(I> ••• • [{ •• O. R)

is a function dependmg anly on

[{i.O

and R.

2. (Bieberbach). Let f(z)=ao+alz+···+a.z.+a.+lz·+ I + ••• be regular with

lai I~

[{i(i=0.1,2.3.···.n). 0.+ 1#0 and assume the values 0.11l() mare than n.m(n~m) time, respecti»ely. m

Iz I
Then R~.Q.~ .. ([{ O. [{I> •••• [{•• a.+1 ). Here .Q.~ .. ([{ o. [{ I> ••• • 1\"

a.+I) is a functl/.J1l dependmg anly on

[(i

and a.+I'

Theorem 1. 4.3 and Theorem]. 4. 4 are obtained by setting m=n= 0 in the above two results. respectively. Furthermore.Corollary 2 of Theorem 1. 4. 1 can be generalized to derive some sufficient conditions of normality for family of meromorphic functions. (Caratheodory.Bloch). Let F= {f.(z)} be a family of functions that are lT1eT'OITlQT'phoc m a domam D and WI. W2 ... • ."wq be a set of a distinct values(may inclue =). Assume that far

(!IffY

functl/.J1l fE F except finilRly many of them. the multiplicities of the roots of the equations fez) = q

Wi (i=

1. 2 •••• • q) in D are 1l() less than

mi.

where the

mi

satisfies i

b=

...L
1UJT'-

I mi

mal in D.

(Bloch). Let F= {f.(z)} be a family of functions that are regular in a dOlllain D. If. to each function fE F. the radius of the circle of univalence of the image feD) is bounded by a con-

stant mdependent of f. then F is normal in D. (Ahlfors). Let F= {f.(z)} be a family of fUnctions that are lT1eT'omarphic in a domain D. If the pre-lJ1I(Jge sets di = f- I (Wi) (i= 1.2.3) of an arbitrary function f E F are all lymg 0utside of D. far three mutually exdusWe simply connected regions WI. W 2 and W 3 on the Riel1wUTl sphere that lies tangientially above the w-plane. then F is normal in D.

THEOR Y OF NORMAL FAMILIES

53

5. The Applications of the Theory of Normal Families Theorem

'.5.

'.(Picard). If fez) is a nonconstant 7T1RT'ommphic fUT/dian, then ii can amii at

most tuu values (possibly including 00).

Proof.

Suppose that fez) omits three values:wl,w2,and

Wa.

Note if necessarily we can

consider Wa-W2

g(z)=--' W3-WI

fez) -WI • f(z)-w2

Thus we may assume, without any loss of generality, that

WI' W2

,and

Wa

are 0,1,00 re-

spectively. Set f.(z)=f(2·z).

We now consider the sequence of functions (f.(z)} in is regular and omits values 0, 1 in {f.(z)} is normal in

Izl<1.

1z 1<

Iz 1<1,

and note that each f.(z)

1. By virtue of Corollary 1 of Theorem 1. 4. 1 ,

Furthermore, since f.(O)=f(O), we have, by Theorem

1. 2. 11, that {f. (z)} is bounded in

1z 1<8

(0<8<1). Hence, in

1z 1 <8,

If.(z) 1 = If(2'z) 1<M.

This shows that fez) is bounded in the entire finite (complex) plane. According to Liouville theorem fez) must be a constant, which is a contradiction.

Theorem D. Then

'.5. 2.(Picard). Let

fez) be a nonconstant 7T1RT'07nurphic functwn defWRd in a dOIT/l1in

f assumes every vahle, wiih tuu poss1Ne exceptinns at most, in any ne"9hJxJrhood of an

/,SO-

lnted essentwl smrruJariiy point of f. In other ward, if f (z) is Tegular for 0< 1z- Zo 1
actuaUy the theorem implies that, f takes each of those T/mleXceptwnal values infinitely many times.

Proof. We may suppose that the essential singularity point is zo= O. Choose To suffiCiently small, so that fez) is meromorphic in 0< 1 z 1
7'0.

z

j.(z)=j("2:)

CHAPTER

54

and consider {f.(z)} in ro: ~
such that If., (zo) I <M for all the k. By Theorem 1. 2. 11, we have, for sufficiently small 0,

for r;

+ 0< Iz I
is bounded in a sequence of ring-shaped

domains that shrinks to the origin. Particularly, f is regular, and hence, is bounded in a domain formed by two such ring-shaped domains: 0< Izl
the same contradiction follows by considering {f)Z)}

I

noting that if f (z) omits 0, I ,

00, so does l/f(z).

Theorem ,. 5. 3.(Julia). Let f (z) be a tran.scendental entire function. Tllen tllere exists at least one direction: Arg z = f{>o suell tllat given any e> 0 , f (z) takes every value, witll tuu possible excep-

tions at most, m tile sector:f{>o-eo is cal1Rd a Julia

d~rectio1l

(of f).

Proof. First we show

that {f.(z)} = {f(2'z)} is not a normal

family in +< Iz I <4.

According to Weierstrass theorem (or Picard theorem above), when k-oo, there exists {z.} with Iz 1_ 00 satisfying If (Zt) 1<1 such that to each z, corresponding an integer n.

satisfying

~<

I ;:, I <2. It follows that n,-oo when k-oo. Now if {f. (z) } is normal,

then a subsequence {f." (z) } of {f., (z) } can be selected so that it converges uniformly to a limit function fo(z) on Iz I ~2. Since

z., If"'(2":)

1<1,

hence fo(z) cannot be a constant. Therefore, {f.,.(z)} is bounded on ;

~ Izl~2.

Con-

sequently, fez) is bounded in 2·'.-I
THEORY OF NORMAL FAMILIES

55

nite plane. This is inconsistent with the assumption that f is a nonconstant function. Thus {f,(z)} cannot be normal in + 0, there are infinitely many functions f,(z) in the sequence {f,(z)} such that each of the functions takes every value, with two common possible exceptions at most, an infinity of times in I

z- Zo I <

o.

It follows that

f takes every value, with two possible exceptions at most, an

infinity of times in Iz- 2'zo I < 6 • 2'. Since 6 can be chosen arbitrarily small, the argu-

ment:Arg zo=rpo is then a Julia direction of f.

Examples of Julin direction 1. Arg z= ±

;

are Julia directions for fez) =e'. Let e>O be given. Then e'takes

every value other than 0, an infinity of times in ; - e
+ e;

to any a (::;t: 0 ,

:;t.oo), z=log a+2n:n-i (a chosen fixed branch of log a) will fall inside ; -e
; +e, when n is sufficiently large. The case Arg z= -

; can be discussed similarly. Moreover, there is no other Julia

direction for e'. This can be shown by the fact that, to any given e> 0, within IArg z I ~

; -e, IArg

z-:n-I~; -e,!!.n;, e'=O,oo

respectively.

2. Arg z= O,:n- are Julia directions for sin z. Given e> 0, sm z takes every finite value an infinity of times in IArg z I <e. In general a meromorphic function f (z) with a period w has Arg z=w,Arg w+:n- as its Julia directions. By Picard theorem fez) takes every value, with two possible exceptions at most, an infinity of times; moreover, if f (zo) =

a, then f(zo±nw) =a. Hence, to any given e>O, when n is sufficiently large, z±nw will fall into IArg z-Arg

wi <e, IArg

z-Arg w-:n-I <e respectively.

3. Every direction is a Julia direction for the

Weierstrass elliptic function:9(z).

Since, to any given rp and e>O, the sector:rp-e
CHAPTER

56

1

Polya showed: The Julia directions of any meromorphic function formed a closed subset of the interval: [0, 2:7rJ and, conversely, given any closed subset S of [0, 2:7rJ , one can construct an entire function of infinite order having S as its set of Julia directions. The set of Julia directions of an infinite order entire function, and the singularities of a function regular in Let

fez)

Iz I< 1 are related

= ~a.z" 11=0

by the following facts:

be an entire function of infinite order and F(z) =

~b.Z"

be

.=0

a Taylor series with radius of convergence equal to 1. Then I . F(z) has at least one singularity on I

I •

f

Iz 1= 1 ;

(z) has at least one Julia direction;

H . (Vivanti-Diense). If Ikg H'. (Polya). If

I kg

a. I~

b.1 ~ ; - 6, 6> 0, then

; - 6, 6> 0,

z= 1 is a singularity;

then the positive x-axis is a Julia direc-

tion;

m.

(Hurwitz-Polya). By

~ e.b.z·,

suitably choosing e.= ± 1, one can construct FI (z) =

having z= 1 as its natural boundary;

11=0

m'.

By suitably choosing e. =

± 1,

one can make fl (z) =

~ e.a.z·

having all the

11=0

directions as its Julia directions; IV. (Fabry). Let N(n) denote the number of nonzero terms in the first n terms of

the sequence:b o,bl>b 2 , · · · . If limN(n)=O, then the set: n boundary of F(z) ;

Izl=l

becomes the natural

IV I . (Polya). Let N(n) denote the number of nonzero terms in the first n terms of

. N(n) the sequence: ao, al , a2 , .... If l~m - - = 0, then every direction is a Julia direction for f 11_00

n

(z). Theorem 1. 5. 3 can also be rephrased as follows. Let fez) be a transcendental entire function. Then given e> 0, there exists a sequence of points {z.} with z.-oo such that in

Iz-z.l<elz.1

O. 5. 1)

fez) takes every value, with two possible exceptions at most, an infinity of times.

Condition 0.5. 1) can also be replaced by assuming

THEORY OF NORMAL FAMILIES

57

Iz-z.I<e.Iz.I,

(1. 5. 2)

e.-O

where {€.} is a sequence properly chosen. Moreover, it is known (Valiron-Milloux) that the rate of decrease of {e.} is related to the increasing property of T (1' ,f). The same conclusion holds for a function fez) which is regular in a neighborhood of an isolated essential singularity zo, or f may not be regular but omits at least one value in a neighborhood of zoo Generally speaking, if there exists a continuous curve L which tends to the isolated essential singularity

Zo

such that

f(z)-a as

Z-Zo

along L, then

Theorem 1. 5. 1 holds.

Theorem 1. 5. 4. Let z= 0 be an isolated essential singularity of the functllJll f(z), which is me1'amorphic

HI

0< 1z 1< 41'0.

0 such 1'40< 1z 1< 41'0, ii

If there exist a value a arul a continuous CU1've L COII1X!T'gmg to

that f(z)-a as z-O along L, then (f.(z)}

=

(f( ;. )} is not normal

UI

follows that there exists at least one di1'ectwn: Arg z = 'Po such that, for any gwen e> 0 , f (z) takes

every value an mfiniiy of times, with two possible exceptwns at 1nost, in -e
that there exists a sequence of points {z.} such that d(f(z,) ,a)~o>O as z.-O. Each z, corresponded to an integer n, satisfying ~ < If (J.(z)} is normal in

can

12"z,1 <21'0.

Set 2·,z,=z,·.

~0
be selected that will converge spherically uniformly on

1'; < 1z 1~ 21'0

to some limit

function f 0 (z). Since f. , (z;>=f(2-·:, )=f(z,),

it follows that d(f. , (z;) ,a)~o, and hence fo(z) cann't be a constant equal to a. Furthermore, to any

l'

satisfying 1'; <1'< 21'0 and p = 1 ,2, •.. , there exist 0, (O~O,

< 2rr)

til

and the corresponding points ~, = 1';.: on L, such that when p-oo,

f., (1'e tll,) = f(~,)-a. The 0,' s has at least one cluster point, say 00 , and f. , (z) converges spherically uniformly to f 0 (z) on 1'; ~ 1z 1~ 21'0 with f 0 (retllo ) = a. Such a 0, exists that corresponds to each

l'

CHAPTER

58

with r;
Thus far, we have only considered families of functions {f.(z)} (;.)} with (in

r

lying in

0< 1z 1 < 4ro case),

~
= {f(2·z)}

(in the whole finite plane case) or

or {f

~o
respectively. However, the same argument works for families of

functions {f.(z)}={f(
0< 1z 1< 4ro.

Particularly, we can consider the

family of functions {f.(z)}={f(2±oe ·z)}. Also let a=a(t)(O
representation of a continuous curve that converges to z=oo or z=O as t-I. By consid-

= {f(a(t)z)}; O
ering the family of functions: {f,(z)} conclusion; here

the case when the singularity

Theorem

'.5.5.

Zo

being equal to

00

as follows.

Assume thnt fez) is reqular on Izl~R, which IIfJ8 z=oo as an /'solated eSC'Rll-

tial smqularity. Let a(t) be a ContinUOllS

CllT'1Je

started from a pomi

= R awl ew1ed at z= co

< z < r2 (R then m the domam: 1z- zoa(t) 1 <e 1a(t) 1 , fez) takes e'l'ery value

as t m()'l}mg from 0 to 1. Now we, in an arbitrary domain: r I 0, choose a suitable poillt zo,

011 1 z 1 1

1

wilh poss1hle two exceptWns at most.

A meromorphic function may fail to have a Julia direction, if it fails to have an asymptotic value; such a function is called Julia exceptional function.

Theorem ,.

5. 6. (Marty). Let

f (z) be merommphic in the fmllR plane. Then a necessary and

suffICient condition for the family of functWIIS: {f. (z) } = {f(2·z)} to be normal m

+< 1z 1<

4 is thnt

1 Df(z) =O( l;r).

Proof. According to Theorem 1. 2. 9 that the following condition: that there exists a con-

2·lf' (2·s) 1 stant M(o) satisfying Df.(s) = 1 If(2.S) 12

+

<

1 111(0)( 4-0< 1

sary and sufficient one for {f. (z)} to be normal in that the assertion of the theorem follows from this.

s I<4 -

+< Iz 1<4.

0) is a neces-

It is easy to see noW

THEORY OF NORMAL FAMILIES

59

Also we can conclude easily that, from the above theorem and the definitions of A

Cr,!) and T(r ,f), any Julia function fez) must satisfy A(r,f) = o (log r), T(r,f)=O((log r)2). Ostrowski obtained a necessary and sufficient condition for a function f to be a Julia exceptional function in terms of the distributions of the zeros and poles of f. First of all, the above growth condition implies that fez) is a meromorphic function of zero order. Hence by Theorem 1. 2. 3 fez) can be expressed as

IJO _

z)

ap

fez) =Z"

IJO-.!...-) b.

Then a necessary and sufficient condition for such a function fez) to be a Julia exceptional function is the following set of conditions: 1°, there exists a constant

[{I

such that

In(r,oo) -nCr, 0) I<[{I J

2°. there exist constants [{ 2 and [{ 3 (independent of r) such that n (2r , 00 ) <[{2,

and n(2r, 0) - n(r, 0)

3°. there exist constants

-

n (r , 0)

<[{3 respectively;

[{4

and [{s such that the following inequalities hold for any p

and q:

4 there exists a positive e, such that, for any ),., Jl, 0.

I ~A -ll~e>O . For example, if

q> 1,

•

then

is a JUlia exceptional function. Similarly, for a given meromorphic function f(z), a necessary and sufficient condition for the family of functions: (f.(z)}

=

(f(z+a.)} with {a.} being any arbitrary se-

qUence to be normal in the entire finite plane is that

CHAPTER

60

1

If' (z)1 , 1+ If(z) 1 2 <1\.

Consequently, any function f (z) having the property mentioned above must

sati~-

T(r ,f) =O(r 2 ).

Functions that satisfy the above condition include elliptic functions, bilinear functions of the exponential function e', rational functions, and Julia exceptional functions, etc. Particularly, for function f, its associated normal family of function {f. (z) } = {f (z+a.)} fails to have constant functions as the limit function for any arbitrary sequence {a. }, one can show further that such f shares similar properties with elliptic functions

and can be regarded as a generalization of an elliptic function. Similar discussions can be applied to sectorial domains and the following results can be obtained.

Theorem

'.5.7. Letf(z) bererJUlar in thedOlTlllin D:a
D. Assume that f is rounded on kg z=r far some r(aO, f (z) is rounded in 0< Izl
Theorem

'.5.8. Let fez) be regular and is euher ollluing the varues 0,1 ar rounded

In

D: a<

kg z
0)0

on

a+e~Arg z~p-e,

far any gwen e>O.

We note that the same conclusion holds if the straight line Arg z=r is replaced by any continuous curve L that lies in

Theorem

a+e~Arg z~p-e(e>O!arbitrary).

'.5.9. Let fez) be merolll&phic in a
some a' ,P' wuh a
limf(re'a' )=WI r-O

If WI::f=W2' then fez) has at least one Julia

d~rectwn:

and

limf(reiP' )=W2 exist. r-O

kg z=r satisfymg a'

~r~p'.

6. Application to Univalent Functions In this section we consider only univalent functions regular in the disk Iz I < L We

use u to denote the set of univalent functions fez) regular in Izl
THEORY OF NORMAL FAMILIES

Theorem ,. 6. ,.

(J'

is

1Ul1'71U1l

61

m 1z 1 < 1.

Proof. Since the arbitrary fez) belonging to

the point Zo given the minimal value of 1f (z)

(J'

is regular in

Izl~1- ~

1on 1z 1= 1- ~

(0)0), then

must satisty (1. 6. 1)

fez) IS , reguI Moreover, as --;;ar'm 1z 1-----1 >:::::: -

mal value on

1z 1= 1- ~.

f~Z)

But since

0 an d 2'

" cannot be zero, w h'ICh ta k es t h e mml-

is equal to

1 at

the origin, If(zo)

1~ 1Zo 1<

1.

= ;go)) ,then o Izl=1-2'.

We consider f* (z) 1f*(z)I~1 on

f* (0) = 0, f* (zo)

= 1,

f* (z)

# 1 (z#zo) , and

We assume the branch of g(z)=--If* (z)-1 such that g(O)=i, then g(z) is regular in

lzl <1- ~.

Let W=f* (z) and W' =g(z), it follows that W'

2+ 1=W, and IW

1<1 corresponds to W' 2 + 1 < 1, which is, in fact, the internal part of the lemniscate 1W' + ~ 11W' - i 1= 1. Since W = f* (z) assumes all values of 1W 1< 1 in 1z 1< 1- ~ , 1

1

when g(O)=i,W' =g(z) assumes all values of the lemniscate containing W' =i. Moreover, since f * (z) is univalent. 9 (z) cannot assume those values of the lemniscate containing W'

=-

i. Selecting a sufficiently small positive number e. then we have: Ig(z)+il~e>O

for all 9 (z) and

1z 1

< 1- ~.

that {g(z)} is normal in

1

Hence, by the Corollary 2 of Theorem 1. 2. 10, we find

z 1 <1-

~

, and g(O) =i is bounded for all the g' s, then, ac-

cording to Theorem 1. 2. 11, {g (z)} is bounded in

1z 1~ 1-

o.

Hence, f* (z)=g(z)2+1, it follows that f(z)=f(zo) f* (z) is also bounded in

Zl~l-o. Since

0 is

arbitrary, {fez)} is normal in

Izl<1.

1

From this proof, we can

immediately obain the following theorem: Theorem ,.

6. 2. Far

all those functions fez) belonging

If(z) 1~
to

(J'.

they satisfy:

0.6. 2)

CHAPTER

62

Theorem 1.6.3. Far all those functions fez) belonging to

If(z) on

(J,

1

they satisfy:

1~q.>2(r).

0.6.3)

Izl =r. Proof. If the inequality 0.6.3) doesn ' t hold, then there exist (f.(z)} and {O,}

such that

If. (re t8. ) I<-.l.. n

Since (f.(z)} is normal in the domain r-o<

Iz I
where f.(z)#-O for n= 1,2 .

••• ,there is a subsequence {f., (z) }C {f. (z)} such that {f., (z)} converges uniformly to fa (z) ,which must be a constant zero by Theorem 1. 2. 3. It follows that {f'., (z)} converges uniformly to fl o(z), i. e. a constant zero. This is a contradiction with f'., (0) = Q.E.D.

1. This theorem shows that the image of fez) on domain

Iz I~r

contains the disk

Iw

1~q.>2(r).

Theorem 1. 6. 4. If a sequence of univalent functions {f. (z)} in dmTUJin D converges spherically unifarmly m general sense to the limiting fUnction fo(z)"¥=constant, then fo(z) is univalent m D.

Proof. If fo(z) is not univalent, then there exist two points (Zl )

= fa (Z2) = (;0.

Zl

and

Z2

such that fo

Since fa (z) is not a constant, there exist two non-overlapping disks

C1 and C2 with centers

Zl

and

Z2

respectively in D such that fo(z) #-(;0 on the circles of C]

and C2 • From the Corollary 2 of Theorem 1. 2. 3 we get that as n sufficiently large, f. (z) can take (;0 on C 1 and C2 • This is in contradiction with f.(z) to be univalent.

Corollary. The limit of an arbitrary umfarmly Gont'eTgent sequence m Proof. Since

(J

(J

18

also in

(J.

contains only functions f (z) that satisty f (0) = 0 and fl (0) = 1, any li[1l-

iting function, therefore, can not be a contant. The corollary follows from this abviou sly. In terms of the abstract sense, we say that

(J

is compact. This fact is useful in studY-

THEORY OF NORMAL FAMILIES

63

ing various properties of u. For example, we have already known, by Theorem

1. 6. 2

that rp(r) satisfies If(z) I~rp(r) in Izl~r, while by the corollary of Theorem 1. 6.4, we find, in fact, that there exist functions fez) with If(z) I =rp(r) , provided we have selected

rp( r)

suitably.

Theorem 1. 6. 5. wt {f (z)} denote a family of functions that are regular in (z)

I < M,

f ( 0) =

0,

(0) = 1. Tlien in Izl
and f'

ing to (fez)} is unwalent

tllere exists a

I z I< 1 1JJIt1i

If

constant P.. sucll that any f (z) belong-

Proof. Suppose that such a Pm does not exist, then we may select a subsequence (f.(z)}

from {f(z)} ,such that it satisfies: f.(z.)=f.(z'.)(=w.),

Since

{fez)}

is normal in

Izl <1,

1

1

Iz.I<-, Iz'.I<-. n n

we may select, according to Theorem

1. 2.10,

a sub-

sequence {f., (z)} that converges uniformly to the regular function fo(z). Obviously, fo (z) satisfies fo(O)=O and f' (z)l~o>Oon

Izl=p.

0(0)=1.

Hence, when O
that w', - O. Thus we obtain:

~

2m

However, when

n>~, P

J

l'l~p

1#0 holds.

It is also obvious

f' o(z) dz= 1. fo(z)

it follows that: _1_ 2::rri

Thus {fl ,,(z) } and

If we assume suitably P>O, then we have Ifo

J

{f., (z) -w"}

f'

l'l=p

.(z)

d~~2

f.(z) -w. ,.~ .

converge uniformly to fo' (z) and fo(z) on Iz 1=

p, respectively, which, however is impossible. This also completes the proof of the theo-

rem. Note that we may take PM = M -.J M2 satisfying the condition of Theorem

V M2 -

1. 6.

1 in the theorem, and there exist functions

5 and assuming some value twice on Iz I = M-

1. Also the function f (z) of the Theorem is not only univalent in Iz I
it is also starlike. By f being starlike in a domain D, it means fez) is univalent in D and if the image fez) of D contains a point w then it also contains all the points tw (O~t~

1). We may also prove the existence of r .. suth that any fE u is starlike in

Iz I
CHAPTER

64 least upper bound R. of such a r. is called the starlike constant.

Furthermore, we may prove the existence of r., such that any fEu is convex in

zl
By fez) being convex in domain D is meant fez) is univalent in D, and if the im-

age fez) of D contains points WI and W2, then it contains also all the points WI+t(W2WI) (O:::;;;t:::;;;l). The least upper bound R. of such a r. is called a convex constant.

The subset

0"

derived from the functions of

0'

added with other extra conditions

yields different kinds of constants. Particularly, we consider those functions f (z) of a having the form: fez) =z+alz+a2z2+ ••• +a.z"+ ....

Let n be fixed and rp. be the least upper bound of all such a

Ia. I ,then

rp.
more than a fixed times say p. Such a theory is the so called the theory of p-valent functions, with its kinds of theorems already known. Hence, for the set of p-valent functions, substantial results may be derived from the universal constants that correspond ta given properties of the functions. As consequences, many theorems of the univalent functions may be derived easily by means of the idea of normality. Though using the method of normal family may prove the existence of various kinds of universal constants, we may, however, find it difficult to derive these values precisely. But in the univalent function theory, it is really of important that we could determine them precisely. We shall stop here for the discussion of the cases of univalent functions. In the fal-

THEORY OF NORMAL FAMILIES

65

lowing, some theorems related to univalence will be described.

< 1 with f (0) =

Theorem 1. 6. 6.(Landau). ut f (z) be an arbitrary functwn requlnr m 1z 1

and f'

(0) =

1.

<

0

Also the ilTiage set of 1z 1 1 wuler f is a closed domam D m the w- pia1lR. Then

the lTiaXHllal value r J of all these values r such that the entirely ctrcumference ;:; p, where p is a positive

1w 1

=r

lws m D is

constant independent of f (z ) •

Proof. If there does not exist such a p as stated in the theorem,

then to each n, there ex-

ists a function f.(z) with f.(O)=O and f'.(0)=1 and such that the interior of the im-

Thus for

< 1 under f. does not intersect either circle: < 1,

1z 1

age set of 1

z1

1w 1

=

1 or circle: 1 w 1 =

i..

Set

a.- fl. () g. z = f.(z)-fl.' then g. (z) does not assume 0,] and

00

< 1.

in 1z 1

Izl<1.

we find that {g.(z)} is normal in

g.(O)

By Corollary I of Theorem 1. 4. 1 ,

However, as n-oo,

= a.- fl·_ oo , - fl.

thus {g.(z)} has a spherically uniformly convergent subsequence {g., (z)}, with its limiting function being constant

00.

From

f. (z ) = fl. + -g.(z-) , o.-fl.

we find that {f., (z)} must be spherically uniformly cnvergent to constant O. But f' " (0) = 1, which is a contradiction. This also completes the proof. The assumption f' (0) = 1 has been used to conclude that any uniformly convergent sequence in {f(z)} cannot have a constant as its limiting function. Thus the above theorem remains valid for any set of functions that do not have any constant as a limiting function.

Theorem 1. 6. 7.(Valiron). Suppose

that {f(z)}is a family of fUnctions

which are regular m 1

CHAPTER

66

z I~ 1 with f ( 0 ) = 0, and that thRy have no constants as their lunuing functions. Then for any

qt,1J-

en e>O, there exists a p(e»O such that any fez) in (f(z)} takes all varues in Iwl
u'lth the exceptinn of those varues in a sector uriih wrtex at w=O, radw.s pee) and wrtex angle 2e in the w- plane.

Proof. If such a pee) does not exist, we may then select a subsequence (f.(z)} from {j 1 (z)}, such that f.(z)::;;t:a., ::;;t:p., as Izl
n

e<rp.= Arg P.- Arg a.<2:7T-e.

If

:7T 2

3

~rp'~2:7T, then from la.-p.l~ Ip.l, we have

1~1~_1_. I ~p a.- • sm e :7T

If rp.< 2 or

3

rp'>2:7T, then I P. Isin e < IP. I Isin

rp. I < Ia. -

P. I·

Hence a.-p. sm e I ~I<-J· In any cases, we have

.

l ~pl~_l a.-. sm e Let () g. z

then g.(z) does not assume 0,1 and

00

=

f.(z)-P. a.-p. '

in Izl<1. Hence, {g.(z)} is normal in Izl
But

Ip.1 1 Ig.(O) I = Ia.-. P 1<-'sm e is bounded. From Theorem 1. 2. 11, {g.} are uniformly bounded in Iz I ~r(r
is convergent uniformly to 0 in Iz I ~r (r< 1), which leads to a contradiction. This completes the proof. Note that this theorem contains Theorem 1. 4. 4. Let f.(z)=ao+alz+···+a.z"+···, al::;;t:O

THEORY OF NORMAL FAMILIES 1z 1

be regular in

67


is regular in

1z l <1

trarily given

e> 0,

and does not assume

ao a= Ra

1

1, then

z+··· and

l - ao

p=~.

.

Therefore, for any arbI-

it follows that:

lal>p(e),lpl>p(e),Laop<2e, with at least one of them valid. Let

Laop=o,

then

1021+ 1p21- 1a- p 12 COS

0=

21allpl

10012+ 11- aoI 2-1 2l aoll-aol

which depends only on ao. Let e be smaller than a particular quantity q(ao) that is merely related to ao, then we have:

implying that:

CHAPTER 2

HP

Space

,. Harmonic and Subharmonic Functions

1. 1. HARMONIC FUNCfIONS Let f(z) be a function regular in the unit disk: 1 z 1 < 1. Define

M,(r,f)= {

}1/, ,O
1 J2~ { -2 If(re lll ) l'dO 7r

0

M(r,f)= max If(re lll ) I. o.;;;q,;;;2~

Definitions. The class of all the functions f (z) that is regular in the unit disk satisfying

.-1

limM, (r ,f)

<

00

is denoted by H'. The class of all the bounded regular functions defined

in the unit disk is denoted by H=. The class of the real-valued functions u(z) that is har-

monic and satisfying M,(r,u)
h'(O
Since

it follows that a necessary and sufficient condition for a regular function

f E H= is that

both the real and imaginary parts of f belong to h'. Obviously, both H' and h' are linear spaces. Moreover, when 0 have

Let

l-r2 P(r,O)= 1-2r cos O+r2

< p< q~

00,

we

69 be the Poisson kernel. Then the following Poisson integral

u(z) =u(relll )

=

1 21r

Jor~ P(r,(}-t)u(ei')dt

(2. L 1)

is used to indicate the relationship between the harmonic function and its boundary values within the unit disk. In (2. L 1) if u(eot) is replaced by an arbitrary continuous function ",(t) with ",(0)

= ¢( 21r), then u(z) remains to be a function harmonic in Iz I< 1 and continuous in Iz

I~1, with

its boundary values satisfying u(eot) =1/J(t).

In general, a function u (z) represented by the following so called Poisson - Stieitjes integral: u(z) =u(relll )

1 J2# = 21r 0 P(r,(}-t)dp,(t),

where p,(l) is of bounded variation on [0,21rJ, is a harmonic function in

(2. L 2)

Izl
Lemma (Helly' s selection principle). It is poSSIble to select fram a unifarmly bounded and unifarmly bounded variation sequence of fUnctians {p.. (t)} an the intenYlZ [a, b J a subsequence (P,o, (t) } which is convergent unifarmly almost everywhere to a function p,(t) of bounded lXfT'iation with respect to tan [a,bJ. Furthf!rmnre, to any continuaus fUnction ",(t) defined an [a,b], then

we have the follmving relatianship among Poissan- StiRlt JeS mtegrals:

.~':o

J.

",(t)dP,o. (t) =

J.

",(t)dp,(t).

Proof. We refer the reader to Chap. 8 of Natason' s book (Theory of Real Variable[29J).

Theorem 2. 1. 1. In the unit disk the follnuring three classes of functians are equivalent to each oth-

er: i. The Poisson-StwUjes integrals J

ii. The difference of tu'Q nannegative harmanic functians J iii. hI.

PrOOf. Ci)~(ii) is obvious. Since each function of bounded variation p,(t) can be ex-

pressed as the difference of two bounded and nondecreasing functions, it follows that every Poisson-Stieltjes integral is a difference of two nonnegative harmonic functions. (ii)~(iii).

Suppose U(Z)=UI(Z)-U2(Z), where UI(Z) and U2(Z) are two nonnega-

CHAPTER

70

2

tive harmonic functions, then

fon

lu(re 08 )

IdO ~ fow ul(re08 )dO+

J:w U2(re08 )dO=2:7f[UI(O)+U2(O)] ,

hence uEh l • U

E hI. Define

p,.(t)=

J~u(re08)dO,

(iii) ~ (i). Suppose it is given that

.

then p,.(O)=O, and, for O=to
t~IIp,.(tt)-p,.(tt-l) I~

J:" lu(re08 ) IdO~c.

Hence {p,.(t)} is a family of functions of uniformly bounded variation. Then by virtue of the lemma, we conclude that there exists a sequence r.-l such that p,.. (t)-p,(t) (n-oo) ,

with p,(t) being of bounded variation on [O,2:7f]. Therefore, 1 2:7f

Jor- P(r,O-t)dp,(t)

]2" P(r,O-t)dp... (t) = lun._00 21n fa- P(r,O-t)u(r.e")dt 1 = lim -2 a_co

1T

0

It

0

= lim u(r.z) = u(z).

The theorem is thus proved. The fact that in the unit disk, every nonnegative harmonic function can be expressed as a Poisson-Stieltjes integral of nondecreasing function p,(t) is also called Herglotz representation theorem. Also to a given u E hI, the corresponding funtion p,(t) of bounded variation is uniquely determined. Moreover, if

J

2"

o P(r,O-t)dp,(t)=O,

then

+ z dp, (t) = Jor" e: e -z where y is a real constant. Using the expansion

iy

(I z I< 1,

z = re 08 ) ,

71 00

e:+z = 1 +2 ~e-i.t;:', £0

we obtain

-z

f

2W

o e""dp,(t) =

•=

0

1

(n=0,±1,±2,·.. ).

As it is well-known that any characteristic function of an arbitrary interval can be approximated (in V norm) by continuous periodic functions, and hence, by trignometric polynomials, it follows that dp, is a zero measure. 1. 2. BOUNDARY BEHAVIORS OF POISSON-STIELTJES INTEGRALS

If u(z) is the Poisson integral of a continuous and integrable function ep(t), then to

any point t=Oo, u(z)-1jJ(Oo) as z_e08o. The same conclusion can be extended to Poisson-Stieltjes integrals. u(z)-p,' (0 0 ) as z_e

Moreover,

08o •

when p, has continuous first derivative,

then

Furthermore, if the condition on p, to be relaxed to assume that

the symmetric derivative D (0 ) =li1 p,

p,(00+t) - p,(00-t) 2t

....~

o

exists, then we can derive the following result.

Theorem

2. '.2.

2) such that p,

Let u(z) be the Poisson-StieUjes inteqral with respect to p, as defuwd by (2. l.

is of bounded varudion, whose symmetric derimtive Dp, (0 0) exists, then the rOOUJl

limit (i. e. the limit when z ap]J1'oaches the boundary along any rooms) lim u(re 08o ) exists, and

.-1

lim u (re 08o ) = Dp, (0 0),

.-1

r

Proof. We may assume, without loss of generality, that 00=0 and let A=Dp,(O). Then u(r) - A= 21n

P(r,t) [dp,(t) - Adt]

1 = 2n {P(r ,t) [p,(t) -

-~

J"

At] }:'"

[p,(t)-AtiP(r,t)dt.

2n ·W ~ The first term on the right-hand side tends to zero as r-l. For 0< <5 ~

at I

ap 2r(l-r 2) l ~(l-2r CQ8 o+r2)2 -0 (r-l).

It I

~ n,

CHAPTER

72

2

Hence u(z)-A-16-O holds for any fixed 6>0, where

16=-2~

t [P(t)-AtJ[~p(r,t)}t [a

Jor [p(t) - 2tp( -t)

___ 1

]

] A ·t -at:P(r,t) dt.

6

7r

To a given e>O, we may choose 6>0 sufficiently small such that 2t Ip(O-p(-t)

I

A ~e (0
This yields

Consequently, when r-1 u(r) -

Q.E.D.

A.

According to the fact that a function of bounded variation is differentiable almost everywhere, we can derive the following two corollaries immediately.

Corollary'. Any fUnction belangmg to hI has radinl limll almost f!Verywhere.

Corollary

2. If

u(z) is the Poisson-8tiRUjes integral with respect to 7jJ( E V), then lim u(re lil )

.-1

=

7jJ «()), almost f!Verywhere.

As a further investigation, we can point out that when z approaches the boundary non tangentially from the inside of the unit disk, then, almost everywhere, u(z)-Dp(Oo).

Also to any function that belongs to the class of bounded regular functions

H~,

its nontan-

gential limit exists almost everywhere. For 0< a< 7r/2, construct the sector with vertex elil , of angle 2a, symmetric with respect to the ray from the origin through elil • Draw the two segments from the origin perpendicular to the boundaries of this sector, and let 8.(0) denote the so formed region.

Theorem

2. ,. 3. If f E H=,

then the radinl limlis lim f (re lil ) ..-1

exist almost everywhere, moreover, if at 0 0

73

eXists, then as

z--elllo in

an crrbit1'crry sectur 8.(0 0 ) (O
it.

Proof. Since h=C hI, it follows immediately from Corollary 1 of Theorem 2. 1. 2 that the radial limits exist almost everywhere. In order to consider the "sectorial" limits, we shall treat functions f(z), which are regular in the circle

Iz- 1 1<1

and it has a limit L as z approaching the origin along the

positive real axis. Let

f.(z)=f( z) (n=l,2 ••• ),

n

then {f.(z)} is uniformly bounded, hence it forms a normal family. It follows that there exists a subsequence of functions {f., (z) }, which converges uniformly to a regular function F(z) in any closed subdomain of the above mentioned circle. Since f.(z)--L for any real z satisfying 0

< z<

2, it will yield F(z)=L. Further-

more, for z in the region:

n cos a Ikg zl~a<2' -2-~ Izl~cos a (the ray kg z=a, that lies inside of the circle:

Iz-ll
(2. 1. 3)

has a length of 2cos a), we

have f,(z)--L uniformly as n--oo. Hence

f(z)--L (z--O, zE {z: Ikg zl~ a}), the theorem is thus proved. If fez) tends to Las z approaching elllo in any given sector 8.(0 0 ) (a
We call fez) having a non-tangential limit L (at point elllo). We can conclude from the above argument that to any function fez) E H= ,its nontangential limits exist almost everywhere (on the boundary of the unit circle).

1. 3. SUBHARMONIC FUNCTIONS

Let D denote a bounded domain,

(J

D its boundary, and

D the closure of

(z) be a real-valued continuous function defined in D (the value

but the function which is identically equal to

00

00

D. Let g

shall also be allowed,

shall be excluded). Suppose that for any

74

CHAPTER

2

arbitrary subdomain B with BC D, and a function U(z) which is harmonic in B and Con_ tinuous on B, if g(z)~U(z)

(zE aB),

implying g(z) ~U (z) in B, then any function assumes the same properties as g(z) described above is called a subharmonic function in D.

Theorem

2. ,. 4.

Let 9 (z) be a real-vahJed continuous functinn defined m a domam D. A 1U'Ces-

sary and sufficient cmulitinn far 9 (z) to be subharmonic in D is that to every Zo ED, twe is a Po

>0

(de[R1Uling on zo) with

Iz -

I< po C

Zo

D (if Zo #- 00) such that the folluu>mg inequail.ty

holds:

while far Zo = 00

(2.1.4)

Proof. We shall prove the case that zo#-oo. The necessity is easy to show. Assume that

the disk Iz-zo I

=

g(z)

(Iz-zol= p),

then

g(z)~U(zo) = 211( = 211(

S:" U(Zo+ pelll)dO

t"

g(zo+ pelll ) dO.

Now let us prove its sufficiency. Let B be an arbitrary domain in D with BCD. Assume that there exists a harmonic function U(z) satisfying

g(z)~U(z)

(zE aB). Suppose that

there exists a certain point z in B satisfying g(z»U(z). Set h(z) =g(z) -U(z),

and assume that on h(z)~O

B,

h(z) attains its maximal value m(>O) on a subset E C

B. Since

for zoEaB, it follows that E C B. Moreover, because E is a closed set, there

exists some Zo in E such that none of its disk like neighborhood can be contained in B completely. Hence there exists a sequence of numbers {p.} with P.>O such that each of the disk: Iz- Zo I
75 ly in E, i. e. , on part of the circumference h(z)<m. It follows that

r-

1 2n Jo g(zo+p.e08 )dO-U(zo) = 21n

fo"' h(zo+ p.e

<

=

m

08

)dO

h(zo) =g(zo)-l/(zo).

This contradicts with (2. 1. 4). Theorem 2. 1. 4 is thus proved.

Example 1. If f (z) is regular in a domain D and p> 0, then 9 (z)

= 1f (z) I' is

a subhar-

monic function in D. In fact, if f (zo)

= O.

then the conclusion holds naturally. If f (zo) =F 0, then f (z)

is regular in a neighborhood of zoo Hence [j(zo)

J'= 21n

f:-

[j(zo+ pe08 ) J'dO

(for sufficiently small p).

Consequently (p sufficiently small).

It follows from Theorem 2. 1. 4 that

1fez) I' is subharmonic in

Example 2. If u(z) is harmonic in D and p;;::1. then lu(z)

D.

I' is subharmonic in

D.

It is obvious when p= 1. If p> 1, then. by Holder's inequality. we have

where

1 1 -+-= 1. p q

Inequality (2. 1. 4) is thus satisfied.

Example 3. If fez) is a regular function defined in a domain D. then g(z) =wg+ If(z)

1

is

a subharmonic function in D. In fact, when If(zo) 1<1. condition (2. 1. 4) holds obviously. When If(zo) then the inequality holds in a neighborhood of Zo: 1z- Zo 1<po Furthermore. log+lf(z)

and note that log 1 f (z)

1

1=

is harmonic there.

loglf(z)l.

I> 1,

CHAPTER

76

2

1. 4. THE CONVEXITY THEOREM OF HARDY In the following we shall consider functions that belong to H2. If fez)

is regular in

Iz I< 1,

= ba.z· (Izl <1), .-0

then, according to Parseval formula from mathematical analysis,

It follows from this that M (r,f) is an increasing function of r. Moreover, a necessary

and sufficient condition for a function fez) to be in H2 is that

~ la.1 2<00.

Also from

.-0

the principle of maximal modulus one can derive that M=(r,f) is an increasing function of r. In general the following results hold:

Theorem

2.'. 5. (Hardy' s

convexity theorem). Let fez) be requlm m

Izl
and O
<00, then (i) M, (r ,f) is a 1U71Hlecreasing function of r J

(ii)logM,(r,f) is a convex function of log r.

In order to prove the theorem, we prove first the following result.

Theorem

2. ,. 6. Let

g(z) be a subharmnnic fUnction defined in Iz I m(r)

= 21n

< 1.

Defme

fa- g(reoQ)dO (0<1'< 1) ,

then m(1') is TWn-decreasing and is a convex function of log 1'.

Proof. Choose

on

Iz I< 1'2

0<1'1<1'2<1.

Let U(z) be a function that is in

such that U(z)=g(z)

(lzl=1'2).

Now since g(z) is subharmonic, it follows that g(z)
Hence

(Iz I<1'2).

Izl<1'2

and continuous

77 Now we proceed to prove the convexity part of the theorem. Let O
f:

mer )~21n

w

Izl
U(re 08 )

Then

(rl~ r~

df)

rz),

(2.1.5)

and the equal sign holds at the two end points. On the other hand, from Green's formula,

fr-{ low U (re =~ fZW

08 )

dO}

au ds=~,

roan

r

where : denotes the normal derivative, ds=r dO the element of arc length, c a constant. on

It follows that the right side of (2.1. 5) assumes the form:

a log r+b.

We say that m(r) is a convex function of log r, meaning that if log r=a log rl+(l-a)log rz (O
then m(r)~

a m(rl)+(l-a)m(rz).

Now from (2.1. 5) and the form of its right-hand side, we have m(r)~

a log r+b

=a

log(r~d-·) +b

=a(a log rl+b)+(l-a)(a log r2+b) =a[21n

lo" U(rle

08

)dO]

+ (I-a) [21n lo" U (rze

08

)dO]

=a m(rl)+(l-a)m(rz).

This also completes the proof of Theorem 2. 1. 6.

We now proceed the proof of Hardy's convexity theorem as follows.

(0. It has been pointed out in Section 1. 3 that if fez) is regular in D, then g(z) = If(z)

I'

is subharmonic in D.

Similarly, if u(z) is harmonic, then lu(z) I'(p~l) is subharmonic. Thus by applying Theorem 2. 1. 6 we can derive that M,(r ,f) is a nondecreasing function of r.

CHAPTER

78

2

(ii). It has been pointed out at the conclusion of Section 1. 3 that, for p>O and f

(z) regular in 0< 1z 1<1, the function g(z) = Iz 1A If(z) I' is subharmonic in 0< 1z 1<:

1. It follows from Theorem 2. 1. 6 that

rAM~(r,f)

is a convex function of log r.

Now given 0
d

M~(rl ,f) =dM~(r2 ,f)

=

k.

Set r

=

r1d-'

(O
then r),M~(r,f)

< k=k'k l -,= {dM~(rl ,f) }'{r~M~(r2,f) }I-, =r),{M~(rl

,f) }, {M~(r2>f) }1-,.

The assertion that log M(r,f) is a convex function of log r follows immediately by taking the logarithm on both sides of the above inequality.

1. 5. SUBORDINATION Let fez) and F(z) be two univalent functions regular inlzl
rna, 1w(z) 1< 1z 1, It follows particularly that in every disk 1z 1
More generally (without the requirement of the univalency), if there exists a function w(z) regular in 1z 1< 1 satisfying 1w(z) 1< 1z 1 such that fez) =F(w(z)), where f and F both are regular in 1z 1<1, then we shall call f subordinate to F and will denote such a relation by f Theorem

<.F.

2.'.7. (Littlewood). Let

fez) and F(z) be regulm in Izl<1. If f<'F, then

M,(r,f)< M,(r,F) (0< p

<=).

Proof. Let G(z) be a function subharmonic in Izl<1.Consider g(z)=G(w(z)) ,where w(z) is regular with Iw(z)I
f:

w

g(re08 ) dO<

f:

w

G(re 08)dO.

(2. L 6)

In fact, let U(z) be harmonic in Izl
1, then G(z)
( Izl
79 Hence g(z)

~

u(z) = U(w(z))

(Izl=r).

Therefore

= 2~

f:"

U (re.o) rlO= 211f

f:"

G(re.o)rlO.

The assertion of the theorem will be obtained by noting the facts that

1f (z)

I', 1F

(z) I' are subharmonic, and then followed by applying inequality (2.1. 6) letting g(re.o) =f(re.o) , G(re.o) =F(re.o).

2. The Basic Structure of

HI'

2. 1. BOUNDARY VALUES

Definition. We shall use N to denote the class of functions fez) that are regular in the unit disk and satisfy

Since the integration ex p{

2~

is no greater than the integration

fa" logO +

{2~

f:" [1+

1 f(re.o) 1 )rlO}

If(re.o) IJ'rlO}+(where p is an arbitrary

positive number), it follows that

weN (p>O). Theorem

2.2. 1.(F. and R. Nevanlinna). A necessary and suff1C!e1Ii conditlnn far a functinn f

(z) reqular in the unit rlisk belongs to the class N is that fez) can be expresser1 as g>(z)

fez)

= 1/J(z) ,

where g> and 1/J are two fUnctians that are regular and bounrled in the unit rlisk. PrOOf. We first prove the sufficiency. We may assume without any loss of the generality

that

CHAPTER

80

2

1
Then

f:~ log+ If(re08 ) IdO~- f:~ logl1/J(re 08 )

1dO.

Hence, by virtue of Jensen's formula, 2111'

where

z"

f:

n

log 11/J(re08 ) IdO= log 11/J(0)

f:

are the zeros of 1/J. It shows clearly that

1+ I~rlog I: 1' w

is an increasing function

log 1 1/J 1dO

of r, and, hence, fEN. Proof of the necessity. Assume f(z):;i:O and tiplicity

m(~O),

then z-.. f(z)-a

(#0)

fE.V. If z=O is a zero of f with mul-

(z-O). Let z,,(n=1,2,3,···) denote the

rest zeros of f with We know that if fez)

0< IZ11~ IZ21~···<1. #0 on the circle Izl =p «1),

F(z)=log{f(z)~

then

IT (p~2_Z'Z))} z-z,

;;. Iz.l
is a function that is regular in

1

z 1 ~p and satisfies

Re{F(z)}= loglf(z)

1 (Izl=p).

Hence, by Poisson's formula,

1 f2 n F(z) =-2 Ioglf(pei') 11' 0

pe"+~

-"'dt+ic. pe -z

1 -it

IT

z .. p(z-z.) 1 f2W pei'+Z
1 pe"+~ 1/Jp(z) =exp{ --2 log+ 1f(pe") 1-,,-'"dt} , 11' pe-z here Iog-x= max ( -log x, 0). Now we choose a sequence of real numbers: {Pi} with P. increasing monotonically 1, such that f(z)#O on

Izl =p••

Let

then clearly

(lzl<1)·

to

81 Note that rp., 1/J. are regular in the unit disk D and satisfy

Irp. (z) I ~ 1, I1/J. (z) I< 1 in

D.

It follows that {rp.}, { 1/J.} are normal families, and, hence, there exists a subsequence {k j

}

of the natural numbers such that when j-DO ••'JOf'fIIl,

rp(z) (lzl~R
rpkj(z) "'/onn/,

1/Jkj (z) -

1/J(z) (lzl~R
It follows that both rp(z) and 1/J(z) are regular in the unit disk D and satisfy

Irp(z) I~ 1,

11j>(z) I ~ 1 in D. Hence

rp(z)

fez)

= 1/J(z)'

The theorem is thus proved. Theorem 2. 2. 2. Let fEN. Then log If (e,e)

I

1IQ7!-

is mfRgralJle (except f (z) =

Proof. Assume that f(z)~O and that

0).

Furthel"1rwre, if f

E H',

then I (e oQ )

E L'.

IE1". Then

I( ) = rp(z) z

tangential limit f (e oQ ) exists alnwst everywhere, and

(I rp(z) I~l, I1/J(z) I~I).

1/J(z)

where rp(z) and 1/J(z) are two bounded and regular functions in the unit disk, having non-tangential limits rp(e oQ )

,

1/J(eoQ ) almost everywhere. Applying Theorem 2. 1. 3 to Fa-

tau's lemma, we obtain

J:" Ilog Irp(e ~lim

inf {-

r-l

It follows from Jensen's theorem that

and, hence log Irp(eoQ )

IE£1;

Particularly, 1/J(e

oQ )

Hence radial limit f (1'e Moreover, if f E Ce oQ )

H',

oQ )

oQ

)

IdO

Jo/"2" log Irp(1'e IdO}.

J:" log Irp(1'e

oQ )

oQ )

IdO

is an increasing function of

1',

a similar conclusion will be held for 1/J.

cann't be zero identically on a set (of 0) of positive measure. exists almost everywhere, and, furthermore, log If (e oQ )

IE£1.

then, by applying Fatou' s lemma, one can readily verify that I

EL'. The theorem also shows that if fEN and f CeoQ )

=0

on a set of positive measure,

then f (z) - O. It also reveals that any function in the class N is uniquely determined by its boundary values on a set of arguments of positive measure.

CHAPTER

82 From the expression f

=.:e., it shows clearly that if fEN, ¢

2

then

is bounded. Thus, a necessary and sufficient condition for a function f E lV is that

Jo Ilog If (re 2"

oll )

I IdB< 00,

where r satisfies

0< r < 1.

2.2. ZEROS

Theorem

2.2. 3. Let

< Iz. I~ IZ2 I:::;:;:....

f be regulur in Iz I<1 with f~O. Let z.

,ZH •••

Then a necessary arul sufficWnt rondition for

be the

ZRTOS

J:" log I f (re

oll )

of f with 0

I dO

to be

bounded is that =

~ (I-lz.1 )<00 . • =1 Proof. Assume that z= 0 is a zero of f with multiplicity m. Then fez) =dz"'+···

(d*O).

The rest zeros of f are denoted by a.(n= 1,2, ••• ) with

0< Ia. I~ Ia21 ~ ....

According

to Jensen's formula,

which is an increasing function of r I if it is bounded by a constant c (finite), then, for any fixed integer k, t

~/og _r_~ ~ log _r_ .=1

la.1

la.1 ~c-log( Idlr").

(r>

lat I)

1_,1<.

Letting r-1, we obtain t

0< Id le-c~ II la.l, .=1 it follows from this that

~ (I-la.1 )<00.

Conversely, by applying Jensen's theorem, we have

II la.l) exP{-21 J2" log If(re ( 1_,1<. 1r

and the assertion of the theorem follows.

0

oll )

IdO}

< Id I,

83 Corollary. The

Zf!I'O

sets of functinns in N

must satisfy

:6 (1-lz. \)<00. Theorem

2. 2. 4. Let aI, a2' "', be a sequence of complex numbers such that 0 < I al I~

Ia2 I~ ••• < 1 and

=

:6 (1- Ia. I) < 00, then the infinite product .=1

a.-~

BCz)=TIhl a.

.=1

}-a.z

converges unifunnly in every rIisk: Izl~R<1 uriLh {a.} bemg the oole atSk. Mareo~V!T',

IBCz) 1<1 CIzl
ana

IBCe08 ) I =1

Zf!I'OS

of BCz) in the unit

almost everywhere an the umt cr,rcle.

Proof. For Izl~R, we have a. _z~I=ICa.+la.lz)(1:-la.I)1 1 a.", a. C1 - a.",)

I l-hl a.

~ 2(1-la. \)

""" Since in

.

:6 (1- Ia.1 )< 00, it follows that the infinite product BCz) converges uniformly

Iz I ~

R

When

<

1 and is regular in

Iz I< 1,

Iz I< 1.

the modulus of any finite product of BCz) is less than

we conclude easily that with

1-R

IBCz) 1<1 and

1.

From this

the radial limits of BCz) exist almost everywhere

IBC(08 ) 1~1. We now show that actually

IBC(08 ) 1=1

almost everywhere.

To any f E H, we have, from Lebesgue's bounded convergence theorem and the monotonicity of MICr,f),

f:" IfCr(08 ) IrIO ~

r

IfC( 08 ) IrIO.

We apply the above result to f= :. ' where

ir

C)hl at - z • z - t= I at 1 - iitz'

B

Since lB. Ce08 ) 1==1, it follows that

B(re08) , J2. IB(e'"' ) IrIt. J2.' o B. (re08 ) rIO ~ 0

CHAPTER

84

2

But ••i/arrttlr

B.Cz) -

CIzl =r).

BCz)

Hence

21r~

1- I

BCe") Idt.

From this and the fact that IBCe08 ) I~l) Calmost everywhere), we conclude that it must be

IBCe08 ) I = 1 almost everywhere,

and the proof is complete.

In general, the expression:

BC ) - .... ft~ a.-z z -~ _=1 a. l-iiaz' where

In

is a non-negative integer with

is called a Blaschke product.

2.3. THE MEANING OF CONVERGING TO THE BOUNDARY VALUES

Theorem

2.2. 5.CF. Riesz).

Any functinn f(z)(?t=O) thnt beWngs to W(p>O) can be decom-

posed as

fez) =B(z)g(z), where B (z) is tke Blaschke product, g Cz)

E H' and hns

1W

zeros in

Iz I< 1.

Similarly, to every fEN, it can be decomposed as f=Bg, where g beWngs to N and does not vanish in

Iz I< 1.

Proof. We may assume that f has infinitely many zeros (otherwise, the conclusion is obvi-

ous). Let

B.(z)=z'"

ir~ t=1

Then for a fixed nand e>O with -e. Hence

at

Izl

a.-z

1-iit z'

fez) g.(z)= B.(z)'

being sufficiently close to 1, we have IB.(z)

1>1

85

M

~(1-E)'· The value of the integration is thus bounded by a quantity which is independent of rand decreases monotonically with r. From the above, by letting e---O, we obtain

f:" Ig.(re

~M

l'dO

08 )

(2.2.1)

(n,r«l) arbitrary).

By virtue of Theorem 2. 2. 4 we have by letting n-oo, ..,f.....I'

g.(z) -

fez) g(z)=8(z)

This shows that g E H' and has no zeros in

(lzl=R<1).

Iz I< 1,

and the theorem is thus proved.

The proof for fEN is similar.

Theorem

2.2.6. If fE H'(O
f:" If(re

~~ Further more

lim T-l

f

2"

08

l'dO=

)

f:" If(e

08

)

(2. 2. 2)

l'dO.

(2.2.3)

If(re 08 )-f(e08 ) l'dO=O.

0

Proof. Firstly we prove the assertion (2. 2. 3) when p= 2.

Since

then

It follows from Fatou' s lemma that

fo2.

If(re 08 )-f(e08 )

12

dO ~ lim inf 1'"'"1

f2"

If(re 08 )-f(pe08 )

12

dO

0

=2n ~ la.12(l-r·)2 . • -1

Hence when r-l,

f:" If(re

08 ) -

f(e 08 )

1 2 dO

-0; this shows that (2. 2. 3) is true and so

does (2. 2. 2) for p= 2. Now consider f E H' (O
Since [g(z) JtE H2, hence

CHAPTER

86

fo~

If(re08 )

- fo" Ig(e

08 )

l'dO~ J:~ l'dO=

2

Ig(re08 ) l'dO

fo" If(e

08 )

l'dO.

Applying Fatou ' s lemma again, we obtain assertion (2. 2. 3). Next by using the lemma below, (2. 2. 3) can be derived from (2. 2. 2) immediately. Lemma ,. [Pt Q be a measurable subset of the real axis and rp. E £p(Q) , O
J,

t

Irp.(x) I'dx -

Irp(x) I'dx <=(n -

=).

Then

Jg Irp.(x)-rp(x) I'dx-O (n -

=).

Proof. To a measurable set QI C Q, we define J.(Q')= J.,.lrp.I'dx, J(QI )

=

J,.

Irpl'dx,

QI =Q_QI. Then J(Q' )~liminf J.(QI ) ~ lim sup J.(QI )

.-00

a_co

~ lim J. (Q) -

lim infJ. (QI )

~J(Q)-J(Q' )=J(Q').

Hence, to any measurable set QI C Q, J.(QI) -

J(QI) (n -

=).

To a given e> 0, we can choose a set QoC Q of finite measure such that J(Qo)<e.

Now choose 0'>0 such that to every measurable set QCQo it will satisfy J(Q)<e. whenever m(Q)<e. By Egorov' s theorem, there exists a set QCQo(with m(Q)
=.

87 Since (n-=),

it follows that, for sufficiently large n,

f

Q

+ 1,

IgJ.-gJl'dx=

IgJ.-gJl'dx+

+J(.Qo) +J.(Q) +J(Q)}

f

Qo

IgJ.-gJl'dx

t IgJ.-gJl'dx~2'{J.(.Qo) + 1,

IgJ.-gJl'dx
Lemma 1 is thus proved, and, hence, Theorem 2. 2. 6 follows.

Corollary. If fE H'Cp>O), then lim r-l

f

2"

1log +

0

IfCre oll ) 1-l.og+ IfCe oll ) lido = O.

The above corollary can be derived directly from Theorem 2. 2. 6 and Lemma 2 be low.

Lemma

2. If

a~O, b~O,O
Proof. It suffices to prove the lemma under the condition: l~b
log x -..::::p ~J.-Cx-1)'

Cx~1).

Put x=a/b, the assertion follows. To l~p<=, let 1 Ilfll,= {-2 1T

f2. 0

1

IfCe oll ) l'dO},=limM,Cr,f) r-l

and to p==, let

IIfll== 1.1<1 sup IfCz) 1=ess suplfCeoll) I, 0<0;8<2" then it is easily seen that H' forms a normed linear space.

I

CHAPTER

88

2

2.4. CANONICAL FACfORIZATION

Theorem

2.2.7. If

fEH', p
1 loqlf(relll ) 1~2:7J'

f2"0 P(r,()-t)loglf(eot ) Idt.

Proof. By Theorem 2.2.5 we may assume f(z)#O inside the unit disk Iz I <1. Thus log If(z) I is harmonic in Izl
1 log If(prelll ) 1= 2:7J'

J2" P(r,()-t)log If(peot ) Idt 0

it follows from the corollary of Theorem 2. 2. 6 that lim p-l

J

2"

P(r,()-t)log+ If(pe") Idt=

J2K P(r,O-t)log+

0

If(e") Idt

0

On the other hand, by Fatou ' s lemma lim p-l

f

2"

P(r,O-t)log-lf(peotldt~

0

fa" P(r,O-t)loq-lf(eot ) Idt. 0

Now by combining the above two results the assertion follows. We now discuss the factorization problem, assuming that f(z);EO, fEH'(p>O). According to Theorem 2.2.2, it follows that f(e lll ) E U and log I!Celli) lEV. Consider the entire function fez) =exp

{-21:7J' f2. ee;:+z log I f(e") Idt} • -z 0

(2.2.1)

According to Theorem 2.2.5, we have f(z)=B(z)g(Z)1 g(z)#O.Furthermore by Theorem 2.2.7, Ig(e lll ) 1= If(e lll ) I (almosteverywhere)and Ig(z)I~IF(z)1

(Izl<

1). By virtue of Corollary 2 of Theorem 2. 1. 2, we have Ig(e lll ) 1= IF(e lll ) I (almost everywhere) Set e il' = Ig(O) g(O) I an d 8() z = eiYg(z) F(z) wh"ICh"IS reguIar"In

Iz 1<1

"f"les 8(0»0 an d an d satlS

0< 18(z) I ~11 18(elll ) 1=1 (almost everywhere). It shows that -log 18 (z) Iis a positive harmonic function and vanishes almost every-

where on the boundary of the unit disk. It follows from the Herglotz representation(Theorem 2.1.1) that -logI8(z) I can

be expressed as the Poisson-Stieltjes integral of a bounded nondecreasing function p (t)

with

pI (t)

=

0 (almost everywhere). It is easily shown that

89

S(Z) =exp

e"+~ } {- Jo ~dp,(t). e -z 2"

(2.2.5)

Combining the above analyses we derive the factorization: f(z) =eiVB(z)S(z)F(z)

and call the function

1 F(z) =ei)'exJ -2

1

r" e::+z log 7/J(t)dt}

(2.2.6)

rr Jo e -z

an au ler function for a H' function, where y is a real number, 7/J(t)~O, log 7/J(t)

E L1 and 7/J(t) E U.

Expression (2. 2. 4) represents an outer-function. We call a function f(z) regular in

Izl
an inner function, if it satisfies

If(z)l
We know that every inner function has a factorization of the form ei)'B(z)S(z) , where B(z) is the Blaschke product, S(z) assumes the form of (2.2.5), with p,(t) being a bounded nondecreasing singular function and p,I (t) = 0 almost everywhere. Functions like S (z) are called singular inner functions.

Theorem

2.2.8. (Normal factorization theorem).

Every JUnct101/ J(z)

E H'

with f(z)~O

has a um,que faclm'izatinn oj the farm: f(z) =B(z)S(z)F(z) , where B(z) is a Blaschke ]fT'oduct, S(z) is a sin{JUl1lr inner functinn, and F(z) is an

outer func-

tum. Canversely, a ]fT'oduct of such fUnctinns B(z)S(z)F(z)EH'.

Proof. From the previous discussion, the first part of the assertions follows. We only need

to prove the converse part. To this end we shall only show that any outer function (2. 2. 6) must belong to H'. By the well-known arithmetic mean and geometric mean inequalitY,we have IF(z)

CD Illeasure

1'<21rr

J:" P(r,O-t) (7/J(t)Ydt(J).

Arithmetic mean and geometric mean inequality ,if j(z)~O and is integrable with respect to a non-negative

d~ and A(J) = Jj(z)dp(z) and B(J) =ez1'{A(logJ)} be the arithmetic mean and geometric mean of j. respec-

tively. Then B(j)';;;;A(J) , the

"=" sign holds Iff f(x)=- constant.

CHAPTER

90

2

Hence

The theorem is thus proved. Similar factorization holds for a function fEN. In this case ,the function F(z) in expression (2.2.6) satisfying

¢(t)~O,

log ¢(t)EV, and we shall call such F(z) an outer

function of class N.

Theorem 2.2.9. Every function fEN wuh

f~O

can be represented in the farm:

8 1 (z) ] fez) =B(z) [ 8 2 (z) FCz),

(2. 2. 7)

where B (z) is a Blnschlre product, 8 1 (z) arul 8 2 (z) are sinqular inner functions, arul F (z) (wiih (¢(t)

= I f(e") I)

is an outer function of class 1'1'. Conversely, every functum hUlmg the form

(2. 2. 7) must belnng to N. Proof. Assume f(z)=e jy B(z)g(z)

(Izl
where gEN with g(z)#O, g(O»O.

Since log Ig(z) I E hI, it can be represented by a Poisson-Stieltjes integral. Namely log Ig(z)

1 ["2#

I = 2n Jo

P(r,O-t)dv(t) ,

where vCt) is of bounded variation. By decomposing vet) into the absolutely continuous and singular parts, the assertion follows. The converse part is an immediate consequence of the observation that every Poisson-Stieitjes integral belongs to hI.

3. HI' Is a Banach Space 3.1. POISSON INTEGRAL AND HI

Theorem 2. 3. ,. Let fez) be a reqular function m Iz I<1. Then a necessary arul sufficient condlr tion for f to be ex'[fT'essed as fez) wuhfPEV,

18

= 2~

J:# Per ,O-t)fP(t)dt

that fEHI arul fP(t)=f(e j , ) (abnost everywhere).

Proof. If a regular function f assumes form (2. 3. 1), then

(2. 3. 1)

91

hence fEHI. Conversely, assume fE HI and set

2~


fo# P(r,O-t)f(e")dt.

We have, to any p with O
1 f2#

f(pz) = 2n

0

P(r,O-t)f(peoQ)dt.

However, by Theorem 2. 2. 6, (p-l) ,

hence (p-l).

It follows that
Theorem

2.3.2 . If

fez) is a functinn reqular inside the unil disk wl1lt its real part being nannega-

twe, then f must belong to H'(p
Proof. We may assume without any loss of generality that f(O)

=

1. From the hypothesis

we see that the range of f (z) lies on the right half-plane. Therefore, f is subordinate to

~ ~: =P(r,O) +iQ(r ,0), where P(r,O) is the Poisson kernel and 2rsmO Q(r ,0) = 1- 2rcosO+r2

is the conjugate Poisson kernel. Since

~ ~: E H',

P(1 ,0) = 0 (0#0), it follows from Theorem 2. 1. 7 that

fo2#

If(reoQ)

<

J

The theorem is thus proved.

2" 0

l'dO<

j'dO Jor"jl+reoQ l-reoQ

IQ(1,B) l'dO
CHAPTER

92

2

3.2. BANACH SPACE

Lemma. If fEH'(O
1:S;;;;2+lIfll,O-r)-+,

r=

Izl·

Proof. We begin by noting that if p#1 ,then the inequality can be derived easily by Theo-

rem

2. 3. 1.

For p= 1, we may apply Theorem

2.2.5

Blaschke product and gE H' with g(z) :;toO. Thus If(z)

II,.

to obtain f= Bg, where B is the

I:S;;;; Ig(z) I and,

hence, Ilfll,= Ilg

Since g(z)#O, one of the branches of (g(z)Y is single-valued and regular in the unit

disk and belongs to HI. We now use the result for the case p= 1 to derive the inequality. Let

Then [{'CU obviously.

.

Theorem 2.3.3. [{' is the closure (in the Urwrm, O
~ ate.t8.

t=O

Proof. For 00, we can, by

Theorem 2.2.6, choose p<1 such that IIfp- fll,<e/21++.

Let 8.(z) denotes the partial sum of the first n+1 terms of the Taylor series expansion of f around z = O. Since 8. (z) - f (z) ( I z

I=

p) uniformly, we have, for n sufficiently

large,

118.- f pll,<e/2 ++. 1

Now by applying the well-known inequality (a+ b)':S;;;;2' (a'+b') , a>O, b>O, O
we obtain immediately the following inequality

118.- fll,<e As to the case

p~

(n sufficiently large).

(2. 3. 2)

1, one also can derive the conclusion (2. 2. 3) by applying

Minkowski inequality. (2.2.3)

shows clearly that the boundary values function f(e.o)

is the closure (in U norm) of polynomials generated by e08 • Finally, we need to prove that [( is a closed set. Let {f. (e 08 ) } be a sequence of functions from [{, which con-

93 verges, in the U norm, to a function t:p(z) E U. According to the lemma, (J.(z)} is uni1 z 1 ~R<

formly bounded on

1. Thus {f. (z)} forms a normal family and, hence, a sub-

sequence {f., (z)} of {f.(z)} can be chosen which converges uniformly to fez) on

Iz 1~

RO, choose lv' sufficiently large such that (n,m:;:::N).

Ilf.-f .. II,<e Further, when m:;:::N, r< 1, we have

M,(r,f- fm)

= lim M,(r,f., •-=

fm) .

Let r- 1, we obtain

Ilf-f.. II,-O Hence Ilf-f.II,-O and, moreover,

t:p(O) = f(e olJ ) (almost everywhere). Corollary. If 1 ~p~=, then H'

f,8

a Banach space.

The validity of the corollary for 1 ~p< = is obvious. For the caes p= =, one can derive that

[{=

is closed by method of the proof of Theorem 2. 3. 3. It follows that

H'~

is

also a Banach space. We note that

11·11, is not

complete metric space.

a norm for O
CHAPTER 3

Vector-Valued Analysis

1. Vector-Valued Functions 1. 1. VECfOR-VALUED BOUNDED VARIATIONS Let E be a Banach space. We denote the real and complex number systems by g;. and q; respectively, and the adjoint space of E by E' , i. e. E' is the set of linear bounded functionals on E.

Definition

3.1.1. A vector-valued function

f(t) on the interval (a,PJ to E is of

(1) weak bounded variation in (a,PJ if rp(f(t)) is of numerical bounded variation

for every

rpE E' ,

(2) bounded variation if supll

~ (J(pJ-f(aJ) 11<= ;

over every choice of a finite number of non-overlapping intervals (a;,p;) in (a,PJ and (3)strong bounded variation if sup

~ IIf(a;) ;

f(a;-l)

11<=,

where all possible finite partitions of (a, PJ are allowed. The two suprema are know as the total and the strong total variations respectively. It is easy to verify that strong bounded variation implies bounded variation and that

bounded variation implies weak bounded variation. Theorem 3. 1. 1. A function f(t) of urok lxmnded variation is of lxmTUled varwtion (but not neces-

VECTOR-VALUED ANALYSIS

95

sar fly of strong bounded variatinn).

Proof. Var {rp(f(t))} is the total variation of rp(f(t)) on (o,/1J. Then for every choice of a finite number of non overlapping intervals (Od/1J in (o,/1J we have


II ~ (J(/1;) ;

M> 0 such that

f(o;)) II<M

for all choices of a finite number of non-overlapping intervals in (0,/1). The second part of the theorem is demonstrated by means of a counter-example. Let

E be the space of all bounded complex-valued functions g( T) on the interval (0, I) with IlglI=sup{ Ig(T) I :O
~

for for

t
1. For any choice of non-overlapping intervals (0;, /1;) in (0, I) we have

(/1J-f(oJ)

11<1. On

II ~ f ;

the other hand,lIfCtI)-f(t2 )1I=1 for any choice of t l ,t2 E(0,

1) with tl :;t:t 2 ; it is apparent that f(t) is not of strong bounded variation.

1. 2. VECTOR-VALUED INTEGRATION Let gz = Definition

(-

00 ,00) and ~ be a complex plane.

3. ,. 2. A vector function f(t) defined on a subset S of gz (or

in E is

C1 )weakly continuous at t=to if lim Irp(f(t)-f(to)) I =0 for each rpEE' , ""0

(2)strongly continuous at t=to if lim IIf(t) - f(t o) II ""0

~) with values

CHAPTER

96

3

The Riemann-Stieltjes integral can be extended to vector-valued functions. Let f(t) be a vector-valued function on the interval [a,p] to E and let get) be a numerically-val_ ued function on the same interval. We denote the subdivision uo=a
together with the point

'T,(t'_I~'T,~t;)

by n and set

Ilnll=max{ It,-t,-II: Definition 3. ,. 3. Let

i=l, .. ·,n}.

.

8 nU,g)

=

~ f( 'T;) (g(t;) -g(t,_I))'

(3. 1. I)

i=l

Then if lim

8n IInll-O

exists in a given topology, we define this limit to be the integral

f:

(3. 1. 2)

f(t)dg(t)

relative to this topology.

.

Definition 3. ,. 4. Let Sn(g,f) =

~ g( 'T,) (f(t.) -

fCt,-I))'

1=1

Then if lim Sn(g,f) exists in a given topology, we define this limit to be the integral IInll-O

f:

(3. 1. 3)

g(t)df(t)

relative to this topology. Theorem 3. ,.

2. If either of the intel]T'als (3.1. 2)

(Jf'

both will exist in thts topology and

J:

f(t)dg(t) =f(t)g(t)

I~-

(3.1. 3) exists in given topology, then

J:

g(t)df(t).

.

Proof. This is an immediate consequence of the following identity and its dual:

.6 f(t,) (g(t,) -gCt,-I)) i=1

= f(P)g(P) - f(a)g(a) -

.

~ get;) U( 'Ti+I) i=O

f( 'T,)) ,

(3.1.1)

VECTOR-VALUED ANALYSIS

97

subdivision of [a,p], that 'f'i~ti~'f'HH and also that

maXI'f'i+I-'f'd~211:7T1I. i

We shall therefore concern ourselves only with the existence of one or the other of the two integrals (3. 1. 2) and (3. 1. 3).

Theorem 3. ,. 3. Suppose that either (1) f(t) is a strongly contmuous vector-valued functwn on [a,/I] u'lih values mE and get) is a numerically-vahled functwn of bounded tYJ1'iatinn on [a,p], (f{'

(2)f(t) is a vector-valued fUnction on [a,p] with values m E of bounded variation alld get) is

a conttnuous numerically-vahled fUnction on [a,p]. Then the mtegrals

J:

t

f(t)dg(t) and

exist m the 1IOT'm topology. Further, if T: E- E

/.S

g(t)df(t)

a closed lmeaT operator on E to a Banach space

F, Lf f(t) E D(T), where D(T) is the domain of T, and T[j(t)] is strongly continuous in case (1)

(f{'

of bounded VaTiation in case (2), then T(

and T(

t t

f(t)dg(t))

g(t)df(t))

=

=

J: J:

T(f(t) )dg(t)

(3. 1. 5)

g(t)d T(f(t))'

(3.1.6)

Proof. For case (1) it is clear that f(t) is uniformly continuous in the norm topology on [a,p], Hence given e>O there exists an 0>0 such that III(t' ) - f(t")

-t" I<0. Consequently if

11:7T11I, 1I:7T211< ~ °

II<e if only

It'

a simple calculation shows that

IIs#1 -Sw211~2eVaT{g(t)}. This proves the existence of the strong integral for case (1). For case (2), 9 (t) is uniformly continuous on [a, p] and, choosing e and

° in the same manner as above, we ob-

tain

for

1 11:7T111, 1I:7T211<"2°. Now VaT {cp(f(t)) } ~VaT {Re(
~4

sup Icp(

~ (f(Pi) -

+ VaT {Im( cp(f(t)) } f(ai))

I

i

With the summation taken over all finite sets of non-overlapping intervals (ai' Pi) in [a,

If]. Hence by Definition 3.1. 1 there will exist an M>O such that VaT {cpC!Cl) }~Mllcpll.

CHAPTER

98

3

As a consequence

lis"

1

-S"

2

11= 0.,11=1 $UP Icp(s"

1

-s" ) I<2Me. 2

This establishes the existence or the strong integral for case (2). Bacause of Theorem 3. 1. 2 both integrals will exist in each case. We turn now to the second part of the theorem. For any

1r

we have T[S,,(f,g)]=S,,(Tf,g) because of the linearity of T. Further

We

have just shown that lim S,,(f ,g) = JfJ f(t)dg(t)

11,,11-0

•

and applying the above result to T(f(t)) instead of f(t) we see that lim T(S,,(f,g))= lim S,,(Tf,g)= JfJ T(f(t))dg(t). 11,,11-0

0,,11-0

a

Since Tis closed ,it follows that J:f(t)dg(t)ED(T) and (3.1. 5) holds. Similarly ,we can obtain (3. 1. 6).

Definition

3. ,. 5. Let

E, F be topological spee and let (E,

b

,p,) be a finite measure

space. A mapping f: E-- F is said to be simple if there are disjoint sets AI, ••• ,A. E and vectors bH

•••

2..:

,b. E F such that

•

b X.

f(t) =

Then for each A E

b

j __ 1

4

(t)b j

J

we define L fdp,=

Lemma

• ~ p,(AnA)b

j•

b ,p,) be a finlie 7/1RaSUT'e space, b and cpE F' U~ have that:

3. ,. ,. Let (E,

ping. Then far each A E

far all tE E.

and let f: E--F be a simple map-

O)cp( LfdP,)= L cp(f)dp" (2) II L fdp,ll< L Ilflldp,.

The proof of lemma 3. L 1 is obvious.

Definition 3. '.6. Let

(E,2..; ,p,)

be a finite measure space.

( 1 ) A mapping f : E-- F is said to be measurable if there esists a sequence of simple

VECTOR-VALUED ANALYSIS

99

mappings f. :E-F which converges to f almost everywhere. (2)A measurable mapping f :E-F is said to be Bochner integrable if there exists a sequence of simple mappings f.: E- F such that

.~'!

t

IIf.- flldll=

In this case we define

f

A

for each

._00

fdll= lim

f

A

O.

f.dll

AE~.

Theorem 3. ,. 4. Let CE,

~

,Il) be a fmite l/U!aS1lT'e space, and let f :E-F be a Bochner mte-

gralJle mapping. Then (1 ) The function gJCf) :E-~

gJC far each

t

fdll) =

t

gJCf)dll

gJE F' and AE ~.

(2)The function

IIfll :E-:?JZ

is inJRgrable and

II far each

is inJRgrable and

t fdllll~ t

Ilflldll

AE ~.

Lemma 3. 1. 1 guarantees that the Bochner integral on the one hand Lemma 3. 1. 1 implies that {

t

t

f dll is well defined. Indeed,

f.dll} is a Cauchy sequence, and on the

other hand Lemma 3. 1. 1 guarantees that the definition of

fA f d~

is independent of the

choice of the sequence {f.}. Finally, from Lemma 3. 1. 1 and the definition of the Bochner integral we can easily obtain the theorem. Theorem 3. ,. 5. If the sequence of Bochner inJRgrable functinns {f.} converges almost everywhere to a limit function f and if there exists a fUled function F, which is lRbesgue mtegrable, such that IIf.Ct) II~FCt) far all n and t, then f is Bochner inJRgrable and lim .-00

for each

AE ~.

J A

f.dll=

J .4

fdll,

CHAPTER

100

3

1. 3. VECTOR-VALUED HOLDER CONDITIONS

Suppose that f(t) is defined on a path C in the complex plane

~,

with range in a Ba-

nach space E, and that f satisfies the condition

(3. 1. 7) for any tl , t2 E C, where a is a constant and 0
Proof. Suppose for any

rpE E' Irp(f(t 1 )

-

f(t2)) I~a It 1 - t2 l p

holds. It follows that sup

I

(t,.t2 lEC'XC'

rp (

f(t 1 )-f(t2))

It2 -t 2 I

1 ___

p

:::::::::,a

< 00.

From the theorem of uniform boundedness we obtain 2 )11·(t t )ECXC}
The converse is clear from the fact that for any rpE E' • Theorem 3.1.7. If vectur-valURd functions f(t) and get) are defined on the path C with range III a Banach algeUra Eo, and if fE VH(p) and

gE VH(v),

then f(t)g(t)

E VH(J..) ,

min(p, v).

Proof. Since IIf(t, )g(t 1 )

-

f(t 2 )g(t2 )

II

II + IIf(t2 )g(t 1 ) - f(t 2 )g (t 2 ) II ~ IIf(t J ) - f(t 2 ) II max { IIg(t) II :tE C} +max{ IIf(t) II :tE C} IIg(t J ) -g(t II,

~ IIf(t 1 )g(t 1 )

-

f(t 2 )g(t 1 )

2 )

the result of the Theorem 3. 1. 7 holds.

where

}.=

VEcTOR- VALUED ANALYSIS J)efinition

101

3. ,. 7. An element x is said to be regular if there is an element x-I,called the

jJlverse of x, such that X-I x=xx-I=I, where I is the unit element.

J,eJ1una

3. ,. 2. If Eo!.S a BarUJch algebra with unit elPment I, then

Ilx- I II
has

a inverse

X-I

e1J(!T'y elPment in thl! open sphere

and =

x-I=I+

2..: (I-x)'.

(3.1.8)

.=1

=

Proof. Since 1+

2..: (I-x)' is absolutely convergent, ,-I =

(I- (I-x» (I

+ 2..: (I-x)") =

=

(I -

2..: (I-x)") (I- (I-x»

=1 •

• =1

Thus (3. 1. 8) holds.

Lemma 3. ,. 3. The regular elPmRnts farm an open set in the Banach alge1:Jra Eo. Proof. We denote the set of regular element by S. We shall show for xoES that all of the elements in the open sphere IIx-xo 11< IIxol II-I also belong to S. For such an x we have I-xxo 1 II =

II (xo-x)xo l 11<1. By the (xxol)-I=I+

II

previous Lemma 3. 1. 2

=

=

• =1

.=1

2..: (I-xxol)"=I+ 2..: «Xo-x)Xo l )" .

It follows that =

=xo-I

+ Xo-I L.J '" « xo-x )Xo-I)" • _=1

Moreover the series shows that IIx-I-xol II:::;;;;IIxo l 11211x- xoll (1- IIx-xoil IIxoll/).

(3.1.9)

Hence we have the following result.

Lemma 3. ,. 4. The inverse X-I is a continuous functinn of x in S. Theorem

3.'.8. If

the inverse r

l

of f in VH(p,) exists as tEC, then rIEVH(p,).

Proof. Lemma 3. 1. 4 gives that (f(t) )-1 is a continuous function of f(t), thus it is a con-

CHAPTER

102

3

tinuous function of t.It follows that (f(t))-I is bounded. The inequality (3.1. 9) Yields _I

11(f(tl ) )

_I

-(f(t2 )) ~

lI(f(t 2)-11I 2I1f(t l )-f(t2)1I

1I~1-lIf(tl)-f(t2)llllf(t2)-111 aalltl-t21~

---=:: 1- a lllf(t l ) -

f(t 2) II

as where al =sup{ II (f(t)) -III :tE C} and there exists e>O such that ale<1/2 as It l -t 1 1

«

o. 1. 4. VECfOR-VALUED REGULAR FUNCfIONS

3. ,. 8. Let f(t) be a, vector function defined on a domain D in !?Jl (or ~) with values in E. If there is an element f' (t) E E such that the difference quotient

Definition

f(t+h~-f(t)

tends weakly (strongly) to f' (t) as h-O,we call f' (t) the weak

(strong) derivative of f(t) at t. We also say that f(t) is weakly (strongly) differentiable at t. If f (t) is weakly (strongly) differentiable at any t in D. We call f weakly (strongly) differentiable in D.

3. ,. 9. f(z) is said to be regular in D if rp(f(z)) is regular in Cauchy's sense for every rpE E' , where range of f (z) is in E. If a vector-valued function f is regular in qJ', then f is called an entire function or said to be entire.

Definition

Lemma

3. ,. 5. Far

any numerically tYJ/:uRd requlur funetinn in the dartlniT! D and any compact subset

S, there is a fm!te quantuy M(f ,8) such that far f!Very choice of z, z+a and z+ fJ in S

l~fJ(~f(z+a)-f(z))aa

fJ1 (f(z+fJ)-f(z))

I~M(f,S).

Proof. Since

-.L._1_+ afJ t-z

1

a(a-/l)(t-z-a)

1 fJ(a- fJ) (t-z- fJ)

1 = (t-z)(t-z-a) (t-z- fJ)'

VECTOR-VALUED ANALYSIS

1 2ni alP{

!

103

r

f(t)at

Jc (t-z)(t-z-a)(t-z-p)

(f(z+a)-f(z))-

~ (J(z+P)-f(Z))}

,

where C is a closed path in D having a positive minimal distance d both from 8 and from the boundary of D, we have that la I

/! {

(J(z+a)-f(z))-

~~ I -"':::2n

r

~(J(z+P)-f(Z))} I

f(t)at

J(' (t-z)(t-z-a) (t-z- P)

I

~2n'TIIJX{ If(t)

I

I

I :tEC} 0-;P0Var(C).

where Var(C) is the length of C.

Theorem strongly

3. 1. 9. If fez) is reqular in D urith range in E, then fez)

d~ff(!T'entiable

t8

strongly continllOUS and

m D, unifarmly with respect to z m any compact subset of D.

Proof. We apply Lemma 3. 1. 5 to the function g>(J(z)). We write

f(z;a,p)=~p{~(J(z+a)- f(z))a-

a

pI (J(z+P)- f(z))}.

The Lemma 3. 1. 5 then asserts that 19>(f(z;a,p))I~A1(g>,f,8)

for every choice of z, z+ a and z+ P in 8. By the theorem of uniform boundedness. This implies the existence of a finite A1 (J , s) such that Iif(z;a,p) 1I~A1(J ,s). If We take the limit as

P- 0, we obtain

1I~(J(z+a)-f(z))-f' (z)II~lalA1(f,s) a for all z and z+a in 8.

Theorem

Q.E.D.

3. 1. 10.( Cauchy). If f (z) is a regulnr vectar-vabJ.ed function on the aomam

D with

values Ln the B-space E, then

J(,f(Z)dZ=O far ev(!T'Y simple closed rectifW/k contour C in D such that the interior of C belongs to D.

CHAPTER

104

3

Proof. For any linear bounded functional rpE E' we have rp( tf(Z)dZ)= L,rp(f(Z))dZ=O,

hence

t

f(z)dz=(}.

Theorem 3. 1. 11. (Cauchy integral formula). ut f (z) be a reqular functwn on the domain D wah values in E. Let C be a closed path in D, the interim of which

/,8

in D, and let Z be such that

arg (t - z) increase,s by 2n when t de,scrWe,s C (positit'f! oriRntatwn). Then

J

f(t)dt f (')(z)=~ 2ni c (t- Z),+1

for n=0,1,2,···.

(3 1 10) • •

Theorem 3.1. 12.(Cauchy-Hadamard). Given the power series 2:,; a, (z- zo)', a, E E.

(3. 1. 11)

,-0

Set

~=lim P

suplla, 11+. Then the series is absolutely COlll'f!T'gent for

Iz-zo I

far

fI_CY.)

Iz- Zo I> p. The

series cont'erges to a requiar function on Iz - Zo I

Vf!T'gence being uniform m eVf!T'y circle of radiw; less than p.

Proof. The classical argument for the convergence of the series applies if we replace abso-

lute values by norms throughout. It remains to show that the function fez)=: 2:,;a,(z, zo)' is regular for Iz-zo I<po Since the series converges in norm in the disc, we have

for each rpE E' and all valued regular function

Iz-zo I<po The series on the right converges to a numericaIlyin IZ-Zo I

tion 3.1. 9. Theorem 3.1. 13. If fez) is a vecInr-valued regular function in 0:S;;;;R 1<

Iz-zo I
VECTOR-VALUED ANALYSIS

105

then +~

f(z)= where

a.

tf

= 2~i

(t) (t -

~a.(z-zo)·

ZO)-·-I dt and C,

(3.1.12) far

instance,

is the C/,rcle

It-zo I =r(R1
Proof·

lIa.II~211r

t

IIf(t) III t-zo 1-.- 1 Idt I

where MT=max {lIf(t) II: tE C, C is the circle ~

It-zo I =r}. Setting r=R 2 -

e, we get

~

II ~a.lz-zolI~2: lIa.II·lz- zo I' = 11=1

II

1

~

for Iz-zo I
~a.(z-zo)· converges

absolutely •

• =1 -I

Similarly,

~

a.(z-zo) Iz-zol>R1+e is also absolutely convergent. =

~

It follows that

a.(z-zo)· converges absolutely for RI +e< Iz-zo I
For any rpE E' we have =

rp(

~

=

a.(z-zo)·)=

~

rp(a.)(z-zo)·,

11=-00

1. While rp(a.) =-2

m

r rp(f(t)) (t-ZO)-·-Idt. Hence

Jc

=

rp(f(z))=

~ rp(a.)(z-zo)·.

Since rp is arbitrary (3. 1. 12) holds.

Q.E.D.

Uniqueness theorem. If fez) and g(z) are regular in D with mlues in E and if f(z.) =g(z.), n= 1,2, ••• , the points {z.} having a limit point in D, then fez) =g(z) in D.

Proof. For any rpE E' by the classical uniqueness theorem we get rp(f(z) )==rp(g(z)) as

zE D.

CHAPTER

106

3

therefore f(z)=g(z)

as

zED

Q.E.D.

By a method similar to the classical method we can obtain the following result. The maximum principle. Let fez) be defined in a dOlT/Din D of the extended plane and on its boundary e, regulm in D and &rangly wntin'Uf.lU8 in DUe. If sup { IIf (z)

II :z E e} = M,

then either

I

fez) II=M or Ilf(z) II<M in D.

Liouville's theorem. If a vector-valued function fez) is regulm in ce' ,and ifllf(z) II is bounded in ce', then

f

(z)

=cowstant element.

Proof. For any rpE E' we have 1~lIrpllllf(z)lI·

Irp(f(z»

is bounded in ce'. The classical Liouville Theorem yields rp(f(z»

It follows that rp(f(z»

-constant. If f(z);Econstant element, then there are points

Zl

,Z2 such that f(zl) =fl ,j

(z2)=f2,where fl=l=12. Hence there is a rpoEE' such that rpO(f(ZI) ) =l=rpo(f(Z2) )

Q.E.D.

which is a contradiction. Definition 3. 1. 10. Let f have an isolated singularity at z=a and let 00

fez)

= 2...;

a.(z-a)·

be its Laurent Expansion about z=a. Then the residue of f at z=a is the coefficient

a_I'

Denote this by res f(a). Methods of proof of the following theorems are the same as in complex analysis. Theorem 3. 1. 14. Let f be a vectm-valued regulm function except for a finite number of pointS

.

{Zj}j_1 m the domain in ce'. Let e be a closed path contained in D- {Zj}7. Then

1. -2 1Tl

1. where n(e ,z) =-2

:Tn

J~ C

Z-Zj

J C

f(z)dz=

~ n(e ,zj)resf(zj)' j-I

is the topoWgical index of ce' u.;o, respect ot Zj'

VECTOR-VALUED ANALYSIS

107

Theorem 3. 1. 15. (1) If f has a pole of arder 1 at Zo then res f(zo) = lim (z-zo)J(z). Z-2:

o

(2)If f has a pole of arder n at zo, then

1 . d·-I (_l)llimd .... -I«z-zo)·f(z)).

resf(zo)

n

%-%0

H

Theorem 3. 1. 16. If fez) =a.. (z-zo)"+a .. +1 (z-ZO) .. +I+ ••• , where atEE far k=m,m

+ 1, "',

and tf a;;1 exists, then f- I (z) exists and has a pole with arder matzo.

Proof. Since fez) (z-zo)'" fez) a.. II
where 0 is sufficiently small. Thus

It follows that

Therefore

where b.. _ 1 ,b.. -

2 , •••

E E.

Corollary. If fez) has a pole of arder matzo, i,e. f( )

z

=

a_.. (z-zo)..

+

a_ ..+1 (z-zo) ..

1

+"',

and a=!. exists, then f- I (z) exists and has a zero of arder m at zoo We obtain this result using the methods of Thearem 3. 1. 16.

1. 5. COMPACTNESS AND COVERGENCE IN THE SPACE OF VECTOR- VALUED REGULAR FUNCfIONS Suppose that D is a domain in'??, and that {f(z)} is the set of vector-valued regular

CHAPTER

108

functions with range in E, where

Definition

zE D. Further suppose

3. 1. 11. For any compact set S in

3

S is any compact set in D.

D if an arbitrary sequence {f.} ( C {f}) has

a strongly uniformly convergent subsequence {f.,} on S, {f} is said to be normal in D. If the set {f} of numerically valued fanctions is uniformly bounded in any compact S in D, then {f} is normal in D. But in the vector-valued case the result may not hold.

For example, if D is the unit disc, we set fl(Z)=(Z,Q,···), f2(Z)=(Q,z,Q,···), fa(z)=(Q,Q,z,Q,···),

with range in 12 , then IIf.(z) II =

Iz I<1

IIf.(z) - f .. (z)

for n= 1,2, •••. But we have

1I=-J2lzl

as

Izl <1

for any integers m=l=-n. Thus {f.} is not normal in D. Theorem 3. 1. 17. If {f} is 1U71'mal in D with range in E, thRn {f} is uniformly bounded on any compact set SinD. Proof. For any

rpE E' from the hypothesis we obtain

{rp(f(z))} is normal in D,hence {q>

( f (z)} is uniformly bounded on any compact set S C D. Therefore {f} is uniformly

bounded by the uniform boundedness theorem. Theorem 3. 1. 18. If the set {f} of vectar-valued regular fUnctions is unifarmly bounded on any compact set SCD, and if D is a simply con1U!Cted domain, thRn {f} is equicontinuous on S.

Proof. Suppose that Zo is any point in D, and that the disc S(zo, 2ro) with the center Zo and

the radius 2ro is in D. There exists a constant M> 0 such that Ilf(z)II~M

as zES(zo,2ro)

for any f E {f} by the uniform boundedness. From Theorem 3. 1. 11 we get IIf' (z)

1 11= 112:7r'.

Jcf(t\2dtll~2M .4:7rro·~=2M c

-z

:7r

ro

ro

for Iz-zol~ro, where C is the circle Iz-zol=2ro. Because S is compact,lIf' (z)11

VECTOR-VALUED ANALYSIS ~Ml(S)

109

as zES, where M1(S) is a constant dependent on S. By the vector-valued

Cauchy theorem we obtain

hence Q.E.D.

Theorem 3. 1. 19. The set {f} of vectur-vabJed regular functions is 1UJ1'mal in a simply connected damam D if arul only if the range of {f} in any compact set SCD is sequentially compad.

Proof. Suppose the range A of {f} in S is sequentially compact. There exists a set {z.} I' such that {z.} = S. For arbitrary {f.} C {f} and Zl in S, there exists a convergent subsequence {fl .• (ZI)}C{f.(ZI)}. Generally, from {ft-l .• (Z)} we can select a convergent subsequence {ft .• (Zt) }; then {f•. .} is uniformly convergent on S. In fact, for any e>O by the compactness of S we get ,

}dIS (Zt

1

J

,o(T e) )-:::>S,

where S(Ztj,o(; e)) is the disc with the center Zt J and the radius 0(; e). Since {f•.• (Zt)} is convergent, J

For arbitrary zES there exists S(ZtJ,o(+e)) such that zES(Z."o(; e)). Since A is sequentially compact, A is bounded. It follows that {f} is uniformly bounded on S. Thus {f} is equicontinuous on S. We infer that 1 IIf•.• (z) - f •.• (zt,) 11
and

1 as Iz-zt I <0(-3e) and m>n>N= max Nt. Therefore ,

1<;j<;,

J

IIf•.• (z)-f ..... (z) 11<0 for zES as m
i. e. {f •.• } is uniformly convergent on S.

CHAPTER

110

3

Next suppose {f} is normal in D. For any compact set SCD, any sequence {z, }CS and any {f.} C {f} we show that {f. (z.)} has a convergent subsequence. In fact, from the completeness of the Banach space and the normality of {f} we can choose {f'I} C {f.} such that {f.,} uniformly converges to fo as zE Sand k-oo.

On the other hand there exists {z",} such that z,., -

Zo

as k-oo, thus

Zo

E S. Since

{f.,} is uniformly convergent, IIf.·, (zo) - fo(zo)

1 II<-Ze

and

1 II f.· , (z., , ) - f.·, (zo) II<-Ze Thus

Ilf.·, (z,.,) -

fo(zo) II~ II f.· ,(z,.,) - f.; (zo)

II + IIf.; (zo) -

fo(zo)

II

<-.le+-.le=e 2 2 Q.E.D.

1. 6. BOUNDARY PROPERTIES OF A CLASS OF VECTOR- VALUED REGULAR

FUNCTIONS IN S (0, 1 ) Theorem 3. 1. 20. Let f (re oll ) be a fUnction on S (0, 1) u'llh values in a complex reflexive Barwch space E. If

fret)

(i. e. F(e"»

is of strongly bounded varintion on the interval 1

fez) = f(re oll ) =-2

s-

1T

-_

(-1T

1+rei (9-') , 1 -rei(9_0 dF (t).

,1TJ,

put

(3.1.13)

Then fez) is a vector-valued requkrr funtion in S(O, 1).

Proof. By Theorem 3. 1. 3,the integral (3.1. 13) is defined and

F'

(t) (strong

derivation) exists almost everywhere (see (9 J) and is Bochner integrable. From the boundary value theorem in Chapter 6 of (l2J for any fPE E' we get 1

fP(f(z» =-2

1T

S·

-.

1 +rei (9-ol

1

,

-rei(9-0fP(F'

(t»dt

is regular, thus f (z) is a vector-valued regular function. Since E is reflexive, (t) =g(t)

F'

(t) exists almost everywhere, it is denoted by g(t), i. e. j"

almost everywhere.

VECTOR-VALUED ANALYSIS

Theorem

3. 1. 21.

111

If 0 is a number such thllt the BocImPr inteqral

J w

-w

.-1

g(O+t)-g(O-t) dt 1 tg-t 2

(3.1.14)

and F' (0) exist, then li11lf(re tO ) exists and

.-1

li11lf(retO )

=F'

(0) +if(O),

where

fun = -~ 2:7r

JW g(O+t) -g(O-t) dt. tg -.Lt -K

2

1+retO 1 to' Proof· Let H.(O) = -re l - r2

Re(H.«()))

= 1 - 2rcas

()+ r

P.(O) ,

2

and 2rsin 0

I11I(H.«()) )

Q.«()).

Put

JW_.g«()-t)Q.(t)dt,

1 f(r'())=2:7r

then

Set rJJo(t)=g«()+t)-g(O-t) ,

2tg -.L t 2

then from (3. 1. 14) we get frist rJJo(t) is Bochner integrable.

,

1 f(r,())-f«()) =-2 :7r

=...!..

2:7r

Let

JK

1 2 . rsm t tg zt rJJo(t)(l-1

-K

SW rJJo(t) -w

-

(l-r )2

2rOO8 t+ r 2)dt dJ,

1- 2rcos t+r 2 •

(3.1.15)

CHAPTER

112

3

h.(t)= 1-2rcos t+r 2 '

Then O
tw

f/J,(t)h,(t)dt-O

as r-1.

limj(r,e) =J(o). ,-1 On the other hand from Theorem 3. 1. 3 we obtain

i. e.

1 -2

7r

fW P~ (O-t)F(t)rlt -#

1 P,(O-t)F(t) = --2 7r

1 =-2 7r

f"

I~w+-217r fW- . P,(O-t)dF(t)

,

P,(O-t)F(t).

-.

Since

J... 27r

where [(,(t) =

fW P~(O-t)F(t)dt=.!...- fW [(,(t/(O+t)~F(O-t)dt, -.

27r

_.l sin t P~ (t) r

17 and p. 35 in

281n

-w

t

for O
02J). CD Let F(O+t) - F(e-t) 2sin t

G,(t)

,;, F(O+t) -F(O-t) F' (0) ( tun 2 ' t = strong I"Imlt ) .

then

SIn

•-0

Let Go(O) =F' (0). Then Ge(t) is strong continuous at t= O.lt follows that

11217r

tw

1 [{,(t) ~ 6";;;1'1<# sup -2 7r + 217r

CD (0

t6

[{,(t)G,(t)rlt-Go(O)

r

Je-.,.)-<-6,6)

II

IIGo(t)-Go(O)lIdt

[{,(t) IIGo(t) -G,(O) IIdt.

Any sequence of Lebesgue-integrable functions Kn which possesses the following propertie..

K.::;;'O.

f"_" K.(x)dz=l,

(ii)

1 2".

(iii)

If I is any open interval about x= 0, then Z,1II ....pIK.(x)

1=0

(Ix I";;;".) ,

It-oo:ll'E:i

we call an approximale identity.

(3.1.16)

VECTOR-VALUED ANALYSIS

113

SiIlce the right of (3. 1. 16) is arbitrarily small, lim21n

1'-1

Because

t.

f·

.

K,(t)Go(t)dt=F' (0).

-11

f(t)H,(O-t)dt

=

t.

f(O-t)H,(t)dt,

limf(reoll ) =F' (0) +ij(O).

Q.E.D.

,-I Theorem

3. 1. 22. Suppose

E is reflexive Banach space. If f(re oll ) is a rounded vectar-vabJed regu-

1I1r functwn with range in E,where z=reoll ES(O,1) , then f(e oll ) exists abTllJ8t everywhere.

Proof· Let 1jJ(z) =

J:

f(z)dz and let

M=sup{ IIf(z)

Then 11¢(zl) -1jJ(Z2) II~M IZI-Z2! for any

II :zE S(O, 1) }. ZI

,Z2 E S(O, 1). It follows that 1jJ(z) is strong-

ly continuous and of strongly bounded variation on C (0, 1) ,where C (0, 1) = {z: Iz ! = 1 }. Since E is reflexive, f (z) exists almost everywhere on C (0,1). For any rpE E' we have rpCf ) =rp(J) as zE C(O, 1) almost everywhere ,i. e. 1/1' (z)

=f

(z) on C(O, 1) al-

most everywhere.

1. 7. VECTOR-VALUED ELLIPTIC FUNCTIONS

3. 1. 12. If D is open in ~ and f is a vector-valued function on D which is regular except for poles, then f is called a vector-valued meromorphic function on D. Definition

Definition 3. 1. 13. If f is a doubly-periodic vector-valued meromorphic function with periods

UJJ

and w2,where Im(wI/W2)*O, then f is called a vector-valued elliptic functions

with periods

WI

and W2 by VEF.

Obviously, the following results hold. Theorem 3. 1. 23. If f E V EF, then the derivative f E VET. Proof. For arbitrary rpEE' we have

rp(j(Z+WI +W2»' =rp(f' (z»

CHAPTER

114

3

and

Thus fl (Z+WI+W2)=f ' (z).

Since singular points of fl and f are same, fl

E V EF.

Q.ED.

Theorem 3. 1. 24. If fEVEF, and f is not a constant eWment, then f has at least

OTU!

pole.

Proof. Let us suppose that f does not have any poles. Then f is entire. It follows that (z)

II

iif

is bounded on the fundamental parellelogram Po, where Po= {2(tl -l)wl/2+ (2t 2 -l)wd2: O:::;:;t j


By the vector-valued Liouville Theorem we get that f is a constant element. We have obtained the desired contradiction. Thus f has at least one pole. Corollary. If two functions f I and f 2 E V EF have thR same poles on thR periodic parallelogram with thR same 'flT'incipal part at poles, then they are same except far a COllstant eWment.

Proof. fl-!2 is entire and is bounded by hypotheses, hence fl- f2 is a constant element.

Theorem

3. 1.25. If

fE VEF ,and r

l exi.sts(i. e. f· f-I =I), then r l E VEF.

Proof. Since r

l (z)f(z)

= f(z)r l (z) =1

= f-I (z+mwl +1l(i)2 )f(Z+l1lQ)1 +1l(i)2) = f(Z+l/'1(i)1 +1l(i)2 )f- I (z+mwl +1l(i)2) , f-I (z) = f-I (Z+mwl +1l(i)2),

where m, n are any integers.

Q.E.D.

Theorem 3. 1. 26. If f E V EF, then thR sum of thR residues of aU poles on thR perwdic parallelogram equaUi thR zero element O.

Proof. Suppose s is the sum of the residues of all poles on Po, then rp(s)

E' . Thus s=O.

= 0 for any

rp in

VECTOR-VALUED ANALYSIS

Theorem

3. 1. 27. Suppose

115

that E is a commutative Banach algeIJra with unit element and Schauder

baSIS {e.}. Suppose further that f, g E VEF have the same zeros aI, ••• ,a. and poles

fll , ... ,fl.

u'llh the same arder s, and that either f-I ar g-I extsts. Then f ,g are the same except far a constant elell7Rnt.

Proof. If f and g are regular at zo, then f, g can be expanded in power series in a neighborhood of

Zo:

f(z)=

~a.(z-zo)·,

(3.1.17)

.=0

(3. 1. 18) where a. ,b. E E for n= 0,1,2, .•.. Let

where aj"), p~.) E ~ for j= 1,2,·" and n= 0,1,2, •••. It follows that f(z)=

~a?)ej+ ~a?)ej(z-zo)+ ~a?)ej(z-zo)2+ •••. i=1

j=1

i=1

Since a power series is absolutely and uniformly convergent in any S(zo ,TI) where O<TI


(3. 1. 17), for any

e> 0 there exists N

such

.

that for n>N IIf(z)-

~ ~a?)ei(z-zo)OIl<e.

0=1 i=1

Becau~e

.

.

~ ~a?)ej(z-zo)o= ~ ~a?)ej(z-zo)O, j=1 i=1

0=1 j=1

Similarly,

where fj(z) and gi(z) are

~a?)(z-zo)O 0-1

and

~P?)(z-zo)'

respectively, and both

0=1

are numerical elliptic functions. In fact, by Theorem 2. 7. 5 in (11] we obtain that for each j there exists fPj EEl such that

CHAPTER

116

3

where L'j is the complementary subspace of ej. Thus rpj(f(z» =rp;(

~ fj(z)e) =rp;(ej)f;(z). j

Since rpj(f(z»

is a numerical elliptic function, so is fj(z).

Similarly, we can get that gj(z) is a numerical elliptic function. From Theorem 5. 12 (in Chapter

m. 5 in

(27]) we obtain for j=2,3,···,

fj(z)=s;!l(z)

where

Sj

are constants. Therefore

Similarly, g(Z) =gl (z) (el

+~ tjej) , j-2 = 2,3, •••.

where gl (z) is the numerical elliptic function, tj are constants for j

Here without loss of generality, we assume f- I exists. Since fl (z) =sgl (z), where s is a constant, g(z)=s-If(z)(el+

~sjej)-I(el+ ~tjej)'

j-2 In a neighborhood of a pole the proof is similar.

Definition

3.1.14. Suppose

j-2

Q.E,D.

fE VEF and A is an arbitrary element in the range of fez).

If the number of roots, counted according to their multiplicities, of the equation fCz)

=}.

is always s, f is called a vector-valued elliptic function of order s.

Theorem

3. 1. 28. Suppose

is of order s, with paIRs

that E

PI , •••

f(z)=C+

,

is a Banach space with Schauder basIS {e.} I, and that f E I'EF

,p, and orders

SI , ••• ,

s, respectwely. Then

~ (But;(Z-Pt)+Bt2.9(Z-Pt)- B2t3 8i" t-I

, where

,

(z-Pt)

+ ... + (-1 )',(B../(st-1)! ).9<,,-2) (Z-Pt) ,

~St=S, ~ Bu =0,0 ,BtjE E for k= 1,2, ... ,q and t-I

!

t-I

are Weierstrass' s functinns (see Chapter I[ • 5 in (27]).

j= 1,2, .•. , t;(z), ,go (z)

VECTOR-VALUED ANALYSIS

117

Proof. From Theorem 3.7.5, we get

~f.(z)e•

f(z)=

• -1

9

= ~ (c.+ ~ (M:)<;;(z- fl.) +bii)9"(z- fl.) - b2i~) 9'" .=1

!

.=1

(z- fl.)

b(')

+ ... + (-1)'. (S.-1) ! 9"(,.-2) (z- fl.)) )e•• h.

Hence 9

fez)

= ~c.e.+ ~ (~bi:)e.<;;(z- fl.) + ~bWe.9"(z- fl.) .=1

"'=1

_=1

_=1

for k=I,2,···,q and j=1,"·,8•.

2. Vector-Valued

Boundary Value Problems

2.1. VECTOR-VALUED CAUCHY TYPE INTEGRALS Definition

3. 2. 1. Let

C be a smooth curve without self-intersection. Suppose tE C and a

circle C Cto , to) is drawn with the center to and the radius to such that there are only two intersection points t' and til on C(to,t)

nCo

_1_ Where I, is the curve C(to,e)

nCo

Considering the integral

J

f(t) dt

2ni a-',t-to ' If lim_1_

..-o2ni

J

a-I,

f (t) dt t-t o

exists, it is called the Cauchy principal value integral of f(t).

Theorem

3.2. 1. Suppose

to, awl that

f is

strongly

that C is a path and to E C, that fCt)

continuous on C. Then lim_l_

r

f(t) dt

....02ni JC-I, t-to

E VEF(p,) in

a neighb(JT'hood of

CHAPTER

118

3

exwts.

Proof. Obviously, _1 211'i

r

f(t) dt

JC-l, t-to

en e (to, e' ) = I" and suppose I" Cl,. We have II r f(t)-f(to)dtll~ r IIf(t)-f(to)lIldtl

exists. Let

Jl,-l,

Jl,-l"

t-to

It-to I

~ 1,-1" alt-tol~-lldtl· The remainder of the proof is easy by the completeness of the Banach space and standard Q.E, D.

methods used m the complex analysis. By the classical methods we can obtain the following result.

Theorem 3.2.2. Suppose thnt fE VH(p) is a functinn of tEe, where e is a closed path. Let <1>(z) =~

r f(t) dt. Then

2m Jc t-z

<1>+ (to) =~

r f(t)t-to - f(t o) dt+ fCto ?In to-b , 2m to-a

<1>- (to) =~

r f(t)t-to - f(to) dt+f(t o?in to-b Jc 2m to-a

2m Jc

211"

(3. 2. 1) f(t o),

where <1>+ (t) and <1>- (t) are the baundray vafues of <1>(z) as z tends to e from the interwr and the exterwr of

e respectively.

The formula (3.2. 1) is called the vector-valued Plemelj formula.

Theorem 3.2. 3. If fE VH(p) as a functinn of tEe, tlien <1>± (t)E VH(p) for p
E VH(p-e) for

p= 1.

Proof. From Theorem 3.2.2 we find that <1>±(t) exists. For any rpEE', rp(<1>±(t))EH (p) for p± (t)

E VH(p) , where H(p) is a set of numerical valued functions

which satisfy the Holder condition, The proof of the case p= 1 is similar.

Theorem 3.2.4. If IE VH(p) on

e,

then

VECTOR-VALUED ANALYSIS 1 (/J(t) =-2' 1J"~

119

J

fe-r) --d-rE VH(II-) c -r-t

for tEe.

The corresponding classical theorem yields Theorem 3. 2. 4.

Lemma

3.2. ,. Suppose

that the vector-valURd functUJTI [{(z,t)

!.8

regular for z in D+ ,which is

the mter1JJT' of e for each fixed tone, thilt k(z,t) is continuous as a functinn of z domaw D

for each fixed tone, and thilt for each fixed z in D,

J

(l)1>(z)=~

c

21J"'

[{(z,t)dt t-z

(2)1>(z)EVH(II-)

UI

the closed

[{ E VH (11-). Then

is regular when zisin D,

as zED,

(3)1>+(t)=~[(t,t)+~ 2

2m

J c

[{(t,-r)d-r -r-t

for tEL.

Proof. By methods similar to those of complex analysis we can obtain that 1>+ (t) exists.

For any rpE E' we have that rp

(1)(z))=~ 2m

J c

rp([{(z,t))dt t-z

is regular in D, thus (/J(z) is a vector-valued regular function. On the other hand, (rp( 1>(t)) + = rp( 1>+ (t))

=

1. 21 rp([{(t,t)) +-2 1J"~

J c

rp([{(t't-r)) d-r (tE e). -r-

Therefore (3) holds. (1) and (2) are obvious.

Note. For each rpE E' , if (rp(1>(t))+ is defined, we may not get that 1>+ (t) is defined.

For example, we take 1>(z) = (1 ,Z,Z2, ••• ,z',"·) in l2' where (aO,al""

,a.'''')El2=l~,

we have rp«(/J(z)) =

~a.z·. Hence

zE S(O, 1). For any (rp(1>(e olJ )))+=

..... 0

rp=

~a.e·oII

11=0

is defined, but 1>+ (eolJ) can not be defined.

2.2. VECfOR-VALUED SINGULAR INTEGRAL EQUATIONS

In this section, we shall deal with vector- valued singular integral equations with range in a Banach algebra Eo.

CHAPTER

120 A(t)y(t) +~

1n

J

K(t, 'f) y( 'r)d'r=j(t) as tE C, 'r-t

3

(3.2.2)

where C consists of a finite number of arc-wise smooth closed curves which do not cross each other; D+ is the inner domain which is on the left of C traced along a positive direction; A(z) is a reguar vector-valued function which is on D+ with values in Eo and satisfies a vector-valued Holder condition on

D. For each fixed wE 15,

K(z,w) is a regular vec-

tor-valued function in D+ and for each fixed zED+, K(z,w) is a regular vector-valued function in D+ and satisfies a vector-valued Holder condition for both z and won

D.

The

function f(t) satisfies a vector-valued Holder condition on C. Under the conditions just mentioned, we shall study the representation of the solution yet) of (3.2.2).

Let B(z)=K(z,z), then B(z) is regular in D+. In fact, for arbitrary rp

qJ

(B(z»

rp

is regular by Hartogs theorem (see[2]). CDThus B(z) is regular in

(B(z» =rp (K(z,z» D+. Since

EE' ,

E H(p,) on 15,

SupposethatA(t)±B(t)#8

B(z) E VH(p,) by Theorem 3. 1. 6. fortEC, thatA(t)+B(t) haszerosal,···,a.. , with

multiplicities A. for k= 1, "',171 respectively, that A(z) -B(z) has zeros Pl'" ,P. with multiplicities

p,j

for j=l,"',n, that (A(t)±B(t»-1 exists for z in

15

except at the ze-

ros of A(z)±B(z), and that both of

dA. -d 1 (A(a.)+B(at»

(3.2.3)

z•

have inverses. Under the above conditions we obtain yet)

CD

=

(A(t)

+B(t) )-1 A(t) (A(t) -

B(t) )-If(t)

A function F(ZI,Z2) defined in a region D of the space O. is said to be a regular analytic function if the

partial derivatives

aaf

%1

and

aaf

exist at every (interior) point of D.

Z2

F. Hartogs was the first to prove that if F(ZI,Z2) is a regular analytic function in a bi-cylinder

1ZI 1~rl'

1z.1 ~r.

that converges absolu tely and uniformly in every closed subregion of (

(

* ).

*)

VECTOR-VALUED ANALYSIS

121

•

+4(A(r)+B(r»-I,6

"'j-I

,6 Bjr(t)Cp •

(3.2.4)

j= 1 r=O

where

(3. 2. 5) for r

= O. 1 ••••• Ilj - 1 and j = 1 ••••• n.

arul Cjr are defined by the following linear equa-

tions: •

,uj-l

~ ~ :~ «A(z) 1 =-4 nt

f aaa ( ~

B(z» (A(z)

+ B(z»

- I Bjr(z)

),=a. ,Cjr

(A(z)-B(z») (A(z)+B(z») -IK(z.r) ),=a i

C

for

(7=

0, I ••••• A'_I and k= 1 •••• • m.

(3.2.6)

The formula (3.2.6) has a solution exactly when (3.9.1) does. We shall only give some of the main steps needed in the proof the result. If equation (3.2.6) has solutions C p

•

then we can let

ji(z) = (A(z) - B(z» (A(z)

{2If K(z.r) _ (A( r) 1n c

r

i. e. y(z) is defined on

• 2,6 ,6 Bjr(z)Cjr } • "j-l

B( r) )-If( r)dr-

z

D.

+ B(Z»-I (3.2.7)

j= 1 r=O

In fact. by (3.2.3) and Theorem 3. 1. 16 we can conclude

that (Ae;;) ±B(z) )-1 has poles of order Ilj at points a, and Pj. and is regular except for points a, and pj. Because (A(t) - B(t) )-Iexists on C. (A(t) - B(t» -I. K(z.t) (A(t)B(t) )-If(t) and K(z.t) (A(t) - B(t) )-1 (t- pj)' E VH(Il) for each fixed z on D+

It is clear that the first term and second term in {

U C.

} in (3.2.7) are regular. Thus ji(z)

is defined for zE D. By Lemma 3. 2. 1 we show that ji+ (t) exists. On the other hand from (3. 2. 3). Theorem 3. 1. 15 and Theorem 3. 1. 16. we get

1

I}

{ I«z.r) r-z (A(r)-B(r»- (r-pj)r

Bp(z)=:;:(,es

T=P )

for r=O, L.···.llj-1 and j=I.···.n, where Djl(t)=(t-pj)~J(A(t)-B(t»-1

and Cp=y(r)(pj)'

CHAPTER

122

3

Considering a,o:/=-pj and setting z= Pi ,j= 1, "',n in

+ B(z) )-1

y(z) = (A(z) - B(z)) (A(z)

(3.2.8) "1- 1

we see that (A(z)+B(z))y(z)

and-2(A(z)-B(z))

~Bj,(z)y(')(P) have identical r=O

expansions to order jlj-1 at points z= Pj' Letting H/z) =B(z)Djl (z),

we compare values between FI (z) =Hj(z)y(z)

and

for the corresponding cases. Since (3.2.3) holds, we get that Hj(z) are regular in a neighborhood U(P) of the points pj for j= 1, ••. ,no From Theorem 3. 2. 2 we infer that

~

2:rn

J [{(z,-r) (A(-r)-B(-r))-If(-r)d-r T-Z ('

is regular in D+. From the condition (3.2.6) we can verify that y(z) is regular in lI(/l,) for j=l,···,n, as is F1(z). Now the expansion to order jlj - 1 at z= pj of F 2 is F 2 (z) =Hj(P)Y(pj)

+ (Hj(z)y(z)) , (z- pj) + ... fJ

Hence (3.2.8) holds. Let z-t, where zED+ and tEL, from Theorem 3.8.2 and f(t) - [A(t) - BCt) J[A(t) = [A(t) - B(t) J[A(t)

+ B(t) J-I B(t) [A(t)-B(t) J-I f(t)

+ B(t) J-I A(t) [A(t) -

B(t) J-I f(t)

we can obtain through simplification yet) = [A(t) - B(t) J-I [f(t) - 2y+ (t) J.

(3.2.9)

Since A(t)-B(t), f(t)-2y+(t)EVHCjl), by Theorem 3.3.2 and Theorem 3.3.3 we obtain that yet) E VH(jl). From(3. 2. 8)we infer

1. [ACz)+B(z)][A(z)-B(z)]-ly(z) =-2 !ITt

J

[(Cz,-r) [AC-r)-B(-r)]-lfCT)dT

('

-r-z

VECTOR-VALUED ANALYSIS

123

Using (3.2.9), Theorem 3. 1. 14 and Theorem 3. 1. 16 yield

1. y(z) =-2 :Tn

-~ :Tn

J o

J

[{t(z,t) [A(t) - B(t) ]-If(t)dt

-z

o

[{(z,t)[A(t)-B(t)]-ly+(t)dt t-z

=_1_ 2:Trt

J

[{(z,t) (t)dt. t-z y

(3.2.10)

C

Now Lemma 3.2.1 gives

1 y+(t)= 21 B(t)y(t) +-2

:Tn

J

[{(t,T)y(-r)d-rk. T-t

o

(3.2.11)

When used in (3.2.9), (3.2.10) and (3.2.11) imply that yet) satisfies equation (3.

2.

n.

2.3. VECTOR-VALUED DOUBLY-PERIODIC RIEMANN BAUNDARY VALUE PROBLEMS

I'DF is the set of all vector-valued doubly-periodic continuous functions on Po (see

Sec. 7) with periods

WI

and

~,

the range of which are in E.

Suppose an operator [{ is defined by the formula [{f=

-b~t) Jf(T)(~(T-t)+~(t))dT' :Tn o

where fCt) ,bCt) E VDFn VH(p) , and ~(t) is the Weierstrass ~-function.

Theorem

3.2.5.

If f,bEVDFnVHCp), then [{fEVDFnVHCp) and II[{fll~~qllbll~lIf

II", where q is a constant and Ilfll.=11Illxllf(t) ~ 'EC

11+

sup '."2 EO o

IIf(t l )-f(t2 )1I

It l -

t21 #

her C . e o!'s a

,w

0<1'.-'2 1';;;1

path m Po which does nd cross the point O.

Proof. Let

U(T-t)=~(T-t)----..Lt' Then U(T-t) T-

is a continuous function of -r-t.

1 ffCT)-fCt) . t dT, and that s IS the length of the arc from the

Suppose that !/J(t) =~ 1rt

o

T-

124

CHAPTER

3

starting point to the point t on Co. Suppose also that So, S are the lengths to the corresponding points to, tl respectively,

= hand SI - So = o'. Finally suppose 20', and that l = t' to U tot" • We obtain

that tl - to tot" are

that the lengths of the arcs t' to

J

f(t) - f(to+h) t-to-h

) -

f(t 2 )

Co

J

tit-L :In

Co

f(t) - f(t o) t-to

and

tit

Then Ilf(t l

I ~lIfllp It - t l

2

1

P,

1110 I ~ Ilfllp(J It-to-h Ip-I as + J It-to Ip-I as ) n I l

~ IIfllpk op- 1 (3 P+2I+P+ 1 )o'p "'" np

~ Ilfllp(3P+21+P+ npko

l)h P

where ko is a constant which satisfies 0 where

1111

-t < ko< IItSI-S21 l

2

1

~ 1. Thus

IIf 11~lIIlllf lip,

is a constant.

We rewrite I I = I'

I'

r

=

+I",

=L nt

;i

where

J

Co-I

f(t o) - f(to+h) at,

t-t o

JCO-I (f(t) - f(to+h))

(t-t~-h

It follows that

Ill' I = I

;i

(f(to) - f(to+h)) Wg

~II

~ ~ IlfllphP Iarg ~,

I,

::

::

/I

~n~O'o IlfllphP where 0'0 is a function of h, tending to 0 as h- 0, hence there exists a constant that

1112

such

VECTOR-VALUED ANALYSIS

125

Similarly, we have

where

7n3

Let

is a constant.

lh(t)=~ lH

f

Co

!(r:)u(r:-t)ar:. Then

1/J;(t)=-~ 1ft

1/Jl is a regular function of t and

J!(r:)u l (r:-t)ar:. Co

For arbitrary e>O there exists 15 (0<15<1) such that

II1/Jl (?t -t,(t o) II ~ 11/11 ••-.L1lIax -

0

1f'ECo

as It-to I <6 and tECo, therefore

1/Jl E

f

Co

,u' (r:-t) liar:, +e

VH(1).

Since

hence

Let

Then

where I'ar(e o ) is the length of Co. Finally, we consider 1/J3(t) =~ 1ft

J

I(t)tar:, then clearly r:-

Co

Combining the above we obtain

Where

126

CHAPTER

3

Q.E.D. We assume the operator R is defined by the formula Rf = a(t)f(t) ,

where a(t)EVDFnVH(p).

Theorem

3.2.6.

If a(t) ,b(t) ,q(t) EVDFnVH(p) ,

a-I existsaTUi qlia- I Ii.llbli.
then

the singular wiRqral eqUlltinn

f

a(t)f(t) + b(t)

1n

f( -r) (C;( -r-t)) +C;(t) )d-r=h(t)

(3. 2. 12)

"0

has a unique sohdwn.

a-I exists, therefore R- exists, As qlla- I Ii.libli.
Proof. Since

I

I

R- I

,

where V=R-[{ and J is the identity operator.

Q. E. D.

Problem : Suppose that Co is a closed path with an nonempty interior, and that Co is in the fundamental parallelogram Po (see Sec. 7) and the point z=O is included in the interior of Co. Let C= {t+l1l
and Eo be a commutative Banach algebra with the unit element I. Functions (])(z) in V DF are found, piecewise regular and with poles of order at most d at z=O and

Z=111
and which satisfy the boundary condition

(3. 2. 13) where G(t), q(t) E VDFnVH(p) , with G(t)=:j::.O, 0 the zero element in Eo. The type of problem (3.2. 13) is denoted by VDR.

Theorem

3.2. 7.

If

2~i

f

(])-(t)C;(t)dt=O,

(3.2. 1,1)

(a+b)-I exist an Co,

(3.2. 15)

Co

h- I

,

VECTOR-VALUED ANALYSIS then VDR (3. 2. 13)

!.8

127

equivalent to the smgulm inteqral equation (3. 2. 12).

Proof. If we let

=

G(t)

(a(t) -b(t)) (a (t) +b(t)) -I,

get) =h(t) (a(t) +b(t) )-1,

1. fJ>(z) =-2

I (fJ>+{t)-fJ>-(t)) (';-(t-z)+';-(z))tlt,

1n

c.

then we get equivalent relations between (3.2. 12) and (3.2. 13). Suppose that a-I exists, q II b II ~ II b II a-III ~

< 1,

Q. E. D.

h, a, b, fJ>- satisfy conditions (3. 2.

H),(3. 2.15) and 1. fJ>(z) =-2 1I"t

If(r) (.;-Ct-z)+';-(z)tlt,

c.

(3.2.16)

where fet) is the solution of (3.2.13). From Theorem 3.2.14 we conclude that VDR(3. 2. 13) has the unique solution(3. 2. 16). Similarly, we can discuss the singular integral equation with parameter )., given by )" a(t)fCt)

+ bCt?

I f( r) (.;-( r-t) +.;-(t) )tlr=h(t).

1I"t

(3.2.17)

c. Obviously, the equation (3. 2. 17) becomes )"Rf-[{f=h. If I.E p(R- 10, where p(RI

I [()

is the resolvent set of R- I [ ( , then the integral equation

e3. 2. 17) has the unique solution f=RCA, R- I IOR-Ih.

Similarly, we can devive the solution of the following VDR fJ>+ (t,),,)

= fJ>- (t,)") G (t,)") +g (t, ),,)

(3.2.18)

with a parameter)", where G(t,),,)

= ()..a(t) -b(t)

()"a(t)+bCt) )-1,

g(t,)") =h(t) ()"a(t) +b(t)

-I.

Here the description is omitted.

2.4. VECTOR-VALUED BOUNDARY VALUE PROBLEMS WITH THE BOUNDARY A STRAIGHT LINE

Suppose that Eo is a Banach algebra , that the vector-valved function fJ>1 (z) may

CHAPTER

128

3

have a pole at the point ao, which is not on the real axis. 1>, (z) is bounded and regular except on a neighborhood of the point ao and satisfies (3.2.19)

on the real axis,

where G, (t) ,G, (t) ,1>"( (t) E DVH(p,), G, (t)#0, G I ' (t) exists on the real axis, and G, (z) is regular on the upper half plane. Without loss of generality, we will assume ao

=-

i and G, (z) has no zero on the upper half plane.

We first study the homogeneous version of the problem (3.2.19) on the real axis.

1>t (t) =1>1 (t)G, (t)

(3.2.20)

We consider the following formal expression as a particular solution of (3.2. 20) += 1>o(z)=I+

J

tz+l --rp(t)dt, t-z

(3. 2. 21)

where rp(t) is an unknown function with rp(t) E VH(p,), and 1 is the unit element. From (3.2.14) we obtain the boundary values (3. 2. 22) where 1>t (t) and 1>0 (t) are boundary values of 1>o(z) as z tends to a point t on the real axis from the upper half plane and the lower half plane respectively. If z=w(z) =

-+zi, t

z

then w maps the real axis onto the circle C. Let us denote points

on the circle by t, and let

- -

tt

-

-

-

zt

rp(t)=!(-t+i) and 1>o(z) =1>0(- zti)'

From (3. 2. 22) we deduce -Ii .- - 2il-l 1>0" (I+i) =I±mrp(t) et+i)2+

JIi+Ti-1 q;(l) dT-I (T+i)2

'T.

(3.2.23)

c

Let - il G, (t) =G, (-I+i)'

(]jo" (l) =1>0" ( - I ! i ) ' - . 2il-l - a(t)=m ct+i)2(I+G(t)), Ii+ :ri-l - . (~+i)2 (I-G(t) )m,

let) =0, (t) - I ,

(3.2.24)

VECTOR-VALUED ANALYSIS

129 bet) =K(t,t).

We replace (3.2.20) by (3.2.24). Thus

- - - -

1

f

a(t)q>(t)+---=-

1He

K(t,T)- - - - '" t q>(t)dT=f(t) on G.

(3.2.25)

T-

Clearly, --

--

-

1--

--

--

-

1--

(3.2.26)

aCt) +b(t) =t- 2iS(t) ,

(3.2.27)

aCt) -bet) =t- 2iD(t), - -

where Set) =

4rrI

-

4rrOl (t) (t+i)2.

- -

(t+i)2 and D(t) =

(ii(t)±b(t»-1 exists, because (G 1(t»-1 exists. 0Iel;) is regular inside G ,since

- - - is G1 (z) is regular in the upper half plane. The only zero of a(z)-b(z)

1 p= 2i.

We write down some obvious results for the solution of the integral equation (3. 2. 25) as follows:

en

(:z(ii(P)±b(P»)-1 exists.

(ii)If q>(t)

E VH(p,) , then --

IJK(t'T)--- t q>(T)dTEVH(p,).

W(t)=-2'

rr,

(iii) If q>(t)

e

T-

E VH(p,) , then :::i.(-) 1 'f' z = 2rri

J ;-t

K(z,t)-(-)dq> T T

(3.2.28)

C !,s

regular insule C and) J;+

(-t)

'f'

=J...

'-(-t) (2ti-l! (I -G 1 (t) (t+i)2

2 mq>

+_1_

2rri

J

Kct, T) -(-)dT-e q> T T.

(3.2.29)

e

Form (3. 2. 25) and (3. 2. 28) we get Q;(t)=C.f(t)-2q;+(t)(ii(t)-b(t)-1 for tEG.

Let K(t,P)lJ- 1(P)

P-;

b' (P)=[{' .CP,P)+k' iCP,P)=k' I+k' 2,

where K' j = 4rr(I -

0 1 CP»

for j = 1 ,2.

(3. 2. 30)

CHAPTER

130

3

Here there are two cases: (j) If

G1 (/1)=-1 and (I-G 1(/1))-1 exist, then g;(t) =a (t)

1 (1)

(a 2 (t)-b 2 (t)

)-1

J[(St'~)l(T)(a)(T)-b(T)-lctr

(a(t)+~(t))-I m

c

T-t

(3.2.31)

+C Bo(t) (a(t) +b(t)) - I ,

where c= 2D(~) (S(/1) _ 2ki/ )-1 m

J

k(/1 ,:;)l(T) ~a(T)) -beT) )-1 dT. T-/1

c

Proof. we replace (3. 2. 28)by (3. 2. 30) . Then

J

¢(z)=~ 21n

_~ 1n

[((z'T)j(r~(a~T-b(T))-ldT T-Z

c

J

[((z, T)(aST) ;-b(:;)) - I dT T-Z

J

c

=~

[((z,T)l(T)_(a(!)-b(T))-lli-

2m

T-Z

c

We let

then

(a(z) +b(z) )a(z) -b(z))) -I¢(Z) -F(-)-2 -

Z

res

([{(z,t)(a(t)-b(t))_ t-~

;;'(/1)

T-PV-·

Hence (3.2.32) where

7J (:)_

o '" -res

([((Z,rf)(a(i)-b(T))-I)_ t-z T=P

Obviously both of the following two conditions should be satisfied in order that the equa tion (3.2.25) has a solution:

VECTOR-VALUED ANALYSIS

131

Condition (i)(3. 2. 32) holds as z=p Condition (ii) the right side of (3.2.32) is regular in the interior of C. Under condition(1) , we let (3.2.32) with

z=p,

we can get

c=,¢(P)

be the constant element. From the equation

c, then let z in the interior tend to t and replace z

by

t in (3. 2. 32). Therefore we obtain the representation of ;jj+ (0 by substituting this into Q. E. D.

(3.2.30) we derive(3. 2. 31). Under condition (ii) if

GI(P)=-I,

we have the following results:

( 1) If

J

K(P, T)J (i)4(a CT) -beT) )-1 dT=O,

-r-p

c

thell the gellRT'al sowtion of the equation (3.2.25) is (3.2.31), where

c is any constant element

(2)lf

then the equation (3. 2. 25) has no sohdion.

From

Ja)

we get rp(t), hence (/>o(z) is derived.

Theorem 3.2.8.lf «(/>O(Z))-Iand «(/>o±(t))-Iexistfar -oo1 (z) =(/>o(z)P(z) , where P(z) is any vector-valued polynomUll w -

z

i.z + .. t

Proof. The Mobius transformation

t= - t~i maps the real axis onto the circle C ,and prob-

lem(3. 2. 20) becomes: (1)1 + (l) =(1)1 (t)G I (t) on C.

Suppose

(3.2.33)


(1)1 (2) «(I)o(z)) - I is a vector-valued regular function on the complex plane by the vector-

valued Cauchy Theorem. Because (1)1 (z)((I)o(z))-l may have a pole at z=oo, (1)1 (z)«(I)o

(z)) -1 =P(z), where P(z) is any vector-valued polynomial in z.

Q. E. D.

CHAPTER

132 Theorem

3.2.9. Under

3

3. 2. 8,

the general solutlOTl of (3.2.

~I + (t) =~- mel (t) +gl (t) on C

(3.2.31)

the s.me hypotheses as in Theurem

19) is

where P(z) is any vector-vahted polynmnial in _

Z

+zi " t

Proof. The problem(3. 2. 19) becomes

after application of the Mobius transformation w= - t!i' where gl =gl ( - t!i)' From (3.2. 34)we get (3.2.35) Hence ~I (z)(~o (z) )-1 is piecewise regular in the whole complex plane except possibly for a pole at

z==.

From the vector-valued Plemelj formula the solution of the problem (3.2.35) can be expressed as

-

-

- + -

~I(Z)(~O(Z))-I=~ fgl(-r)(;!o 2m

-I

_(-r))

-r-z

c

d:f+P(z) ,

where P(z) is a vector-valued polynomial in z We get the required result for E, t from

Z, t recpectively.

Q. E. D.

If there is a mapping tl =u(t), which maps the upper half-plane into the interior of

Co in Sec. 10,and if its inverse mapping is an elliptic function t=v(t l ), then VDR (3.2.

13) becomes !J>+ (u(t)) =!J>- (u(t) )G(u(t)) +g(u(t)) on the real axis.

Theorem

3.2. 10. If

G(z) is regulur in the interior of Co and if (!J>o(z) )-Iand (!J>± (t) )-I ex-

ist, then any solutm of VDR (3.2.13) run be expressed as: !J>(t;) = vet;)

~i!J>o (v(t;))

2m

J

Co

g( -rl) (!J>ii( v( -rl)) -I vi (-r~) d-rl (v(-rl)-v(t;))(v(-rl)-t)

Proof. We obtain the required result by substituting

l;,,'tl

for

Z,'t

+ P( vet;) )!J>o (v(t;)). in Theorem 3.2.9 respec-

tively, where z=v(t;) , -r=V(-rI), 4>(t;)=!J>I(V(t;)) and g(-rl)=gl(v(-rl)'

Q. E. D.

VECTOR-VALUED ANALYSIS

133

2.5. THE SOLUTION OF THE VECTOR-VALUED DISTURBANCE PROBLEM

Definition

3. 2. 2. If

r/);c ,r/)± ,G, g, g.,.1.

E DVH

(p,) for t E Co, the following vector-val-

ued doubly-periodic Riemann boundary value problem

(3.2.36) is called a disturbance of

(3. 2. 37) where DVH(p,)=VDFnVH(p,) ,lIg.-gll.-O, 11.1.11.-0 as n-oo, and the vector-valued doubly-periodic piecewise regular functions r/).(z) satisfy (3.2.36) on Co which have at most simple poles at z= 0 and

Z='11Ul)1 +00)2'

From the vector-valued Pl'emelj formula and the properties of Weierstrass's <;"-function we can obtain the following result.

Proposition

3. 2. ,. Suppose 1. 1/J(z) =-2 1I"t

where f

f

c.

f(t) (<;"(t-z)+<;"(z))dt,

E DVH (p,), then 1. 1/J±(t)=± 21 f(t)+-2 1rt

f

f(t) (<;"(t-t) +S-(T))dt.

(3.2.38)

Co

For given G,J.,g.EDVH(p,) in (3.2.36), let 1
f

CO

«G(t) +.1. (t) )r/);;;._I (t) +g. (t)) (<;"(t- z) +<;"(z) )dt

(3.2.39)

for k = 0, 1 ,2, ••.. Set r/);;;o(t) =(). We obtain

r/)•. HI (z) -r/)•.• (z) 1. =-2 1I"t

f

(G(t) +.1.(t)) (r/);;;,(t)

-r/);;;t-l (t)) (<;"(t-z) +<;"(t) )dt.

C.

From Theorem 3. 2. 5 and Proposition 3. 2. 1, we obtain 11r/);;;HI-r/);;;tll.~qIlG+.1.II.IIr/);;;.-r/);;;t-lll. ~ (qIlG+ .1.II.)'IIr/);;;III •.

From (3.2.39) we get

1 9.(t)+-2' 1 r/);;;I(t)=--2 11"&

f

C.

g.(t)(<;"(T-t)+<;"(t))dt.

(3.2.40)

CHAPTER

134

3

Since {g.} is bounded in DVH(/l).

(3.2.41) where

1112

is a constant. If

(3.2.42)

.-=

by (3. 2. 40) ltm cJ>;;:.(t) exists for any tE Co and n= 1.2 •.••• and is denoted by cJ>; (I) , thus

(3.2.43) Similarly. 1IcJ>~.-cJ>tllp-O

as k-oo.

(3.2.44)

Equation (3.2.39) and the vector-valued Plemelj formula give

(3.2.45) Therefore 1. cJ>.(z) =-2 Jrt

f

«G(t)+Ll.(t))cJ>; (t)+g.(t)) «(;(t-z)+(;(z))dt

Co

satisfies the boundary value problem (3.2.36). We therefore obtain the following conclusion.

Proposition

3.2. 2. If qIIG+Ll.llp
then the disturbance prOOlem (3.2.36) of the vector-ml-

ued doublypel'iodic RiRlTIl1nn boundary value prOOlem (3. 2. 37) wllh at most a paIR at z= 0 of order

1 has a unique solutwn. which

t8

obtained by the above step-by-step method.

Now we are ready find the relation between the solution of the vector-valued singular integral equation a.(t)y.(t) +b.(t) -. Jrt

f

y.( -r)«(;( -r-t) +(;(t) )d-r= J.(t)

(3.2.46)

Co

and the solution of the disturbance problem (3.2.36) .where a•• b•• y•• f.EDVH(/l) for n =1.2 ... •. Proposition 3. 2. 3. If

(3.2.47)

VECTOR-VALUED ANALYSIS

135

exl-sts, tlWIl the solutwn of the mteqral equation (3. 2. 46) and the solutwn of the disturbance pOO-

tem (3. 2. 36) with the pole al z= 0 of order at

1. -2 1I't

J

llI.ost

1 are equu.'aient under the cmulitwn

rr>; (Ot;(Odt= O.

Co

Proof. Suppose

g.= f.(a.+b. )-1,

(3. 2. 48)

.1.= -G- 2b.(a.+b.)-I. If y,(t) is the solution of integral equation (3.2.46), set

1 rr>.(z) =-2

1I't

J

y.(t)(t;(t-z)+t;(z))dt.

(3.2.49)

Co

From (3.2.49) and Proposition 3.2.1 we get rr>: (0 - rr>; (0 = y. (0,

(3. 2. 50) The integral equation (3.2.46) becomes rr>: (t) - rr>; (t) =rr>; (t) (G(t)

+.1. (t)) +g. (t).

The remainder of the proof is easy, by using the basic properties of the t;-function.

Q.E.D. Suppose operators A., A are defined by the formula A.y=a.(t)y(t)

J +-. J + b.(O 1I't

Ay=a(Oy(O

bet) 1I't

y( 1') (t;( 1'-0 +t;(t) )d1',

Co

y( 1') (t;( 1'-0 +t;(t) )d1',

Co

where a"a,b.,b,yEDVH(p,). Obviously A., A are bounded linear operators on the space DVH(p,). For an arbitrarily small positive number r, set H.= {(a. ,b.): IIA.II~r}.

Proposition

3.2. 4. If IIa.-all~-+O, IIb.-bll~-+O, IIf.- fll~-+O

as n-+oo

(3.2.51)

and there extst r, N such thnt (a.,b.)EH.for n>N, then the following results hold:

(3. 2. 52)

CHAPTER

136

3

(i)The integral equation (3.2.46) has a unique sohdion y•• (ii ) The integral equation a(t)y(t)

bet) +-. 1n

f

y( r) (~( r-t) +~(t) )dr= f(t)

Co

has a unUJUe solution y. (iii)

Ily.-yllp-O as n-oo.

Proof. Since (0 and (ii) are obvious by Theorem 2. 11. 6 in [11 J, we only prove (iii). We have that

II(A.-A)yll

= II (a.(t) -a(t) )y(t) + b.(t) ~b(t) 1n

f

y( r)

(~( r-t) -~(t))dtll

Co

~ Ila. -allpllyllp+cllb. -blip IIYII~, IIA.-AII~lIa.-allp+cllb.-bllp

(3.2.53)

where c is a constant. On the other hand,

A;l- A-I =A;l (A- A.)A- l

,

and IIA;11l is bounded. By (3.2.51) and (3.2.53) we have

IIA;I_A-111-0 as n-oo, i. e.

Proposition

3.2.5. If

conditIOnS (3.2.11), (3.2.52), (3.2.50) and (3.2.53) hold and 112b.(a+b.)-I+GII~-0

as n-oo,

(3.2.54)

then lim t1>.(z)=t1>(z) = hm ~ 11-00

11_00

1ft

f

('0

Y.( r)

(~( r-z) +~(z) )dt

is the solution of the baundary value '[I1'oblem (3. 2. 55) and

t1>(z)=~ 1n

f y(r)(~(r-z)+~(z))dr.

(3.2.55)

Co

Proof. From Proposition 3. 2. 4, we obtain that the integral equation (3. 2. 46) has unique solution. By Proposition 3. 2. 3, the disturbance problem (3. 2. 36) has a unique solution t1>.(z). From Proposition 3.2.4 and (3.2.49) we conclude that (3.2.55) holds

VECfOR-VALUED ANALYSIS

137

and satisfies the boundary value problem (3.2.40).

Q.E.D.

2.6. THE VECfOR-VALUED BOUNDARY VALUE PROBLEM IN 1p

In 1979, J. K. b.J solved the aaubly- perUxlic Riemann baunrIary value '[1T'oblem wiih the rlomain ana the range in a complex plane '6" (see [25J). In 1988,C. G. Hu extended above problem to vector valued cases with the domain in '6" and the range in a Banach space and got some unique solution of this problem under stronger conditions (see [19J). In this section, under weaker conditions we obtain general sol utions (of above problem) belonging to a Banach space U. Let S = {w: Iw I

< l} and S = {w: Iwi:::;;; l}. Set

U = {f:f is regular in S and continuous on S}

with the norm IIfllu=sup{ If(w) I :wES} and the multiplication f

* g=fog. Obviously,

U is a complete Banach algebra.

Suppose that VDF is the set of all vector-valued doubly periodic continuous functions with periods

WI

and

~,

and ranges are elements in U, that DVH(p,) is the set of all vector

valued doubly- periodic functions which satisfy the Holder condition, that 10 is a closed path and has a non-empty interior, that the track 10 of 10 is in the fundamental parallelogram

and the point z= 0 is included in the interior of 10 , that 1= {t+mwl +nw2 :tE 10 , m and n are arbitrary integers} ,

and that the inner domain of 10 is denoted by S+ and S- =Po- (S+

Uzo).

Functions tP(z) in VDF are found to be piecewise regular and with poles of order at most

a at

z= 0 and Z=11iW1 +nw2 and which satisfy the boundary condition tP+ (t) =tP- (t)

* G(t) +g(t) as

where tP±(t),g(t)EDVH(p,) and G(t)EDVH(l). Let

tE 1,

(3.2.56)

CHAPTER

138

3

00

G(t,w)=

~a.(t)w· as

.-0

(3.2.57)

(t,w)Elri XS,

and

Lemma

3. 2. 2. Suppose

thnt 00

~ lIa.III1(l)<='

(3.2.58)

.=0

and that IG(t,w) I<M(M is a canstant) as (t,w)Elri XS.

(3.2.59)

Then the fol1owmg results haM: (i) G(t,w) (i. e. GID(t»

E H( 1)

for each fi:ced

wE S,

tlmt

/,S

GID(t) satisfymg the

L~ps­

chitz condition. (ii) G(t,w) (i. e. G,(w»

E U for each fi:ced tE lri.

(iii) If

Re(G(t,w»=;t:O as (t,w)Elri XS,

(3.2.60)

then (a) mf{ IG(t,w)

I : (t,w) E lri

-

X S} >0

1

(b)Let k(w)=-2 (arg G(t,w»/ (see [28J), then k(w)=k, a constant. (jv) f(w)=

f

1f

•

G(t,w)rltEU.

I.

(v) Let G. (t,w) =G(t,W)/l-l(t) , then

1. L(z,w)=-2 1ft

J

log G. (t,w) t;(t-z)rltEU

I.

for each fi:ced z as z E lri , where

1 1 1 1 /let) =u(t)u(t-2'wl-2'CU:!)/U(t-2'w1 )u(t-2'CU:!) ' u(t) and t;(t) are Wewrstrass u and t; functions respectively. (vi) L + (t,w)

E U for each fi:ced tE lri •

(3.2.6])

VECTOR-VALUED ANALYSIS

139

~

Proof.

CO Since

G.. (t) =

~ Ea.(t)w·,

.=0

~

I<~

IG.. (t))-G .. (t2)

.-0

la.(t l )-a.(t2)

Ilw"1

~

< .-0 ~ lIa.lIl1(I) It l -t2 1·

(3. 2. 62)

(ii)G,(w)EU, because =

~

IG,(w) (iii) (a)If

I<~ a=O

la.(t)

I<~ Ila.lIl1(l)· .=0

(3. 2. 63)

(3. 2. 61) does not hold, then there exists a double sequence {( t. , w.. ) }c

lo X S (n, m = 1 ,2, •.. ) such that t.-to and W.. -Wo as nand m-oo respectively and lim lun

(it,

(wm ) =0. From (3.2.62) we obtain

IIG',-G,.llu-O as n-oo

(3.2.64)

and {G,(w)} is normal on S, hence {G,(w)} is equicontinuous on S. Therefore for any e

> O.

there exist 1\7 and M such that

Thus

IG" (w.. ) -G,. (wo) I
It

It

1

0

(wo)

I

1

<2e+2e=e as n> ~V and

m> M. Therefore G (to, wo) = O.

This is in contradicting with (3. 2. 60).

Hence (3. 2. 61) holds. (b) From (3.2.62) we show that G.. (t) is a rectifiable curve for each fixed win S. Now we can show k(w) is bounded in S.In fact, if k(w) isn't bounded, then without loss of generality, we may assume that there exists a sequence {w.} such that 1

-2 (arg G(t,w.)), 1r

-00

and W.-Wo as n-oo.

•

Since {G, (w)} is equicontinuous,

IIG... -G"olI-O as n-oo,

(3.2.65)

CHAPTER

140

where

IIGw ll=8Up{ IGw(t) I :tElt}. From

the Lemma 3.2.2 (iii) (a) and (3.2.65)

3 We

can conclude

1 1 -2 (ary G(t,w.))/ 0 --2 (ary G(t,wo))/0 as 10 10

11- 00 ,

(3.2.66)

hence

1 -2 (ary G(t,wo))/ 0 =00. 10 It follows that there exists a to E It such that G (to, wo)

= O. This

yields a contradiction,

hence k(w) must be bounded. If there exist two points wSIl and wS 2 ) E S such that

Let WI

= ~

(wS Il +wS 2 ») , without loss of generality, we can assume

Similarly let

W2

=i-(wS Il +WI)' without loss of generality, we can assume

1 1 -2 (ary G(t,wS Il ))/0 ¥--2 (ary G(t,w.))/0 , 10 10 where w.=

~

(wS Il +w._I)' Thus there exists an integer N such that 1 (ary G(t,w.))/ =-2 1 (ary G(t,Wb ll ))/ as n>N, -2 10 0 10 0

1 (ary G (t, w. ) ) 1 are integers. This yields a contradiction, hence k (w) = k, a because -2 10

0

constant. (iv) Obviously j(w) E u. (v) From Lemma (iv) we have L(z,w)EU for each fixed zEPo-lt by the condition (3.2.59). (vi) From the Plemelj' s formula in [28J we can obtain 1. L+(t,w)= 21 log G. (t,w)+-2 lOt

f

10

log G.

Cr,w)~('f-t)d'f

VECTOR-VALUED ANALYSIS

1. = 211.og G. (t,W)+-2

J

+~

2m

where w,.,. =

1li(t)1

10

+

nt

141

J'0

(log G. (T,W»

L.:' ( 1 +_1_+ T-;t)rlT m,_ r-t-u'm.. U'm,. U'm .•

log G. (T,W) -log G. (t,w) rlt+~l.og G. (t,w) T-t 2m

li(t)2,

m and n are integers,

J ~, 10

(3.2.67)

T-t

L.:' is denoted the summation except the

term of m=n=O.From the Lemma 3.2.2 (jv) we obtain L+(t,w)EU, because each term in (3.2.66) belongs to U. Corollary (0. Suppose that

1. G. (w) =-2 lTt

J

log G. (t,w)rlt,

10

that X(z,w) = { whEre II( z, w)

Ut(Z)h(Z'W)eL("Wl as zES+,

(3.2.68)

h(z,w)eL("wl as zES-

= u(z) / u(z-G. (w»,

anrl that

Enpo=0

(3.2.69)

where E is the range of G. (w), then the follou.mg results are valul: (a)X(z,w)EU for each fixed zEst. (b )x+ (t,w)

E U for each fixed tE lo • ~

Corollary (li). Suppose thal g(t,w) = L.:b.(t)w·, aruI that .=0

(3. 2. 70)

g,(w)EU aruI gw(t)EDVH(p.) as (t,w)Elo XS, then

J 10

g(t,w) X+( ) (':;-(t-z)+':;-(z»rltEU aszElo. t,w

We use above Lemma 3. 2. 2 and similar method in [25J easily getting the following results. Theorem 3.2. ". Suppose that G(t,w) 8UtisfiRs canditums in Lemma 3.2.2 aruI (3.2.69), that g(t,w) satisfiRs the canditwn (3.2.70), we have the jollou.i,lIg cases: (OIf G. (w)

=1li(t)1 +710)2' u>here

m and n are integers and

711 2 +n 2 ::;i:

O.

(a)Suppose k+d>O, then a general solution of (3.2.56) can be expressed as

142

CHAPTER

cJ>(z,w)=X(z,~,) 2m

3

f

g(t,w) (s(t-z)+s(z))dt ,.X+(t,w)

+ X(z,w) (ao(w) +al (w)s' (z)

+ ... +aHl-1 (W)S(Hl-1) (z)) ,

where aj(w)(j=O,l,···,k+d-1) are arburary in U mul cJ>(z,w)EU far each fw:ed

zE

st. (b)Suppose k+d=O, then the general sohdinns of (3.2.56) uruier the canduion _1_ 21fi

are m the farm: cJ>(z,w)=

f

X(z,w) 2' 1ft

'.

f

g(t,w) dt= 0 ,.X+(t,w)

(3.2.71)

g(t,w) X+(t )s(t-z)dt+a(w)X(z,w) , ,w

(3.2.72)

where a ( w) is arburary in U. (c) Suppose k+d=-l arui (3.2.71)

nique arui cJ>(z,w)=

f

X(z,w) 2'

1.8

satisfied, then the solution of (3.2.56) is u-

g(t,w) X+(t ) (s(T-z)-s(t))dT.

,w (d) Suppose that k+d<-l,arui that (3.2.56) is satisfted arui 1ft

1 -2'

f

1ft

'.

(3. 2. 73)

'.

g(t,w) (1)() 0 X+(t )s t dt= asA=l,"',-k-d-l, ,w

then the solution of (3.2.56) is (3.2.73). (ii) Suppose that G. (w) #UMI +7W)2' where m arui n= 0, ±

1 ,± 2,,,', that Go(w) =

G. (w)(uwd WI ,W2) arui Go(w)#O,arui that the range of Go(w) is in Po: (a)Suppose k+d+1>O, then a general solutlDl! of (3.2.56) can be expressed

f'0 ;;.~~w))

cJ>(z,w)=X<;'~')

,w

1f£

(s(t-z)-s(t-Go(w))+s(z)

-s(Go(w)))dt+X(z,w)(al(w)(s' (z)-s' (Go(w)))+'" +aHl(w) (S(Hl) (z) -S(Hl) (Go(w))). where a. (w),'" ,aHl(w) are arburary elements m U.

(b) Suppose k +d

+ 1=

0 arui (3. 2. 71) is satisfted, then (3. 2. 56) lll.ls the unique solu-

tion cJ>(z,w)=

X(z,w) 2' 1ft

f

'0

g(t,w) ( X+(t )(s t-z)-C;(t-Go(w)))dt.

,w

(3.2.74)

(c) Suppose k+d= - 2 arui

1

21fi

r

g(t,w)

J10 X+(t,w) (s(t-Go(w))-s(t))dt=O

(3.2. 75)

VECTOR-VALUED ANALYSIS alld

143

(3.2.71) is satisfiRd then (3.2.56) has the unique sohdian (3.2.74). Cd) Suppose k+d<-2 awl (3.2.61)

~

2:rn

ts

g(t,w) ~(j)(t)dt=O Jlro X+ (t,w)

satisfiRd and

as j=1,".,-k-d-2,

then (3.2.56) has the unique sohdian (3.2.74).

Theorem

3.2.12.

If ranges of

(/)±

(t) ,G(t) and get) are in ll, besules wndiJ,ians in Theorem

3.2. 11, then ranges of solutions of (3.2.56) are in U, and sohdlOUS can be represented by formulas

HI

Theorem 3. 2. 11 , under the sense of isometrical tsomorphtsl1l.

Proof. Let

few)

= ~ a.w· as wE S .=0

and

v = {f : .=0 ~ Ia.1 < with the norm

00 }

IIfll,.= •~ la.l . =0

Obviously, this give a unique one-to-one correspondence between an element (ao, aI, •.. ,a.,'" )of

II and a function few) of V, i. e. V is isometrically isomorphic to ll' Be-

cause I'CU ,conclusions of Theorem 3.2.12 hold.

Q.E.D.

Suppose a Banach space E with a Schauder basis {e.} (see [23J) which satisfies the following conditions: (j)If

xE ~a.e.EE, then _=1

(x)(a)=

~a:e.E E .=)

for any fixed

a~ 1.

(ii) If y=

~b.e.EE, _=1

then (x)(y)=

~a.b.EE . ... 1

Clearly, E is a commutative Banach algebra (see [11 J Chapter iv).

144

CHAPTER

3

Theorem 3. 2. , 3. Suppose vector-valued daubly-perWdic pieceu'l.S'e requlm fuoction ?jJ(z) , except for smqulm paint z= 0 and

Z=71kl)1

+7Ut)2 at most, satISfies tile baundary COTIdibnn

(?jJ+(t))(')=(r(t)}(')

wllere ranges of ?jJ± (t) are twn

In

tile space

II

In

* G(t)+g(t)

lo ,ranges of G(t) and get) are

(3.2.76)

for tElt, In

ll' and

*

expresses tile co1!~"{)II1-

,G(t) E DVH (1) ,get) E DVH (p,) , and tlley satISfy mrUJUs corresponding

COI/-

ditions in Tlleorem 3.2.11 ,tllen tile correspooding solutions of (3.2.76) are

and =

wllere (]>(z)

=

b
Proof. By Theorem 3. 1. 6 for any j we have

I (
2)

We use again Theorem 3. 1. 6 to obtain ?jJ± (t) E DVH(p,/p).

The proof of the rest of this theorem is clear. Corollary. If ?jJ( z) satisfiRs tile conditions of Theorem 3. 2. 13 with the ranges of ?jJi (t)

In

a

H!lhert space,tllen tile baundary value problem (3.2.76) have oorUJUs corresponding solution in U, under tile sense of isometrical isam.orphism, wllere

Proof. Let p=2 in Theorem 3.2.13, we can derive the conclusions as required immediately.

Q.E.D.

VECTOR-VALUED ANALYSIS

145

3. The Analysis of Locally Convex Spaces 3.1. G-DlFFERENTlABlLITY

Definition 3. 3. ,. A set C' (xo) is called a C-star about Xo if C' (xo) =xo+ H, where hE H,

I;:; I~1

implies that

zhE H.

.

Definition 3. 3. 2. A subset D of a real (or complex) linear system E is said to be finitely open if for each choice of elements YI , ••• ,y. E E, the elements

b a.y. which are in D carI

respond to an open subset of the space R. or Z. with elements (aH''', a.). Let p(x,h)=sup{p: Izl~p implies x+zhED}, where D is finitely open. Definition 3. 3. 3. Let y = f (x) be defined in the finitely open set D in values in a Banach space and suppose that for every xE D and hE E the quotient f(x+zh) - j(x)

z which is defined for 0< Izl
(2)of(x,h)=0!f=lim f (x+zh:- f (x) is the first variation of f(x) with respect to z-O

H

the increment h. Theorem 3.3. '.If j(x) arui9(X) rrre G-differentiablR in D, then

o! (af + P9) = Proof.

(af(x+zh)

ao!f

+ P9(x+zh)) -

00 1,8

+ PO!9. (af(x)

af+P9 arui

(3. 3. 1)

+ P9(X))

z =a f(x+zh)- f(x)

+ p9(X+zh) -9(x)

z

z

gives (3. 3. 1). Theorem 3.3.2. o(x,h) is homogeneous of degree one in h, i. e. of(x,ah) =aof(x,h).

(3.3. 2)

CHAPTER

146

3

Proof. Let T)=az. We have f(x+zah) - f(x)

a

z

f(x+T)h)- f(x) , T)

where T)=za. It follows that (3. 3. 2) holds.

Definition 3. 3. 4. A function f (x) defined on E, with values in F is said to be homogeneous of degree n if f(ax) =a'f(x).

Such a function is called bounded if there exists an

M> 0 such that

Ilf(x)II~Mllxllo, for all xEE.

Definition 3.3.5. Let y=f(x) on E with values in F be defined in the finitely open set D. We say that f (x) is F -differentiable and possesses a total or Frechet differential in D,

if (i)of(x ,h) exists as a bounded homogeneous function of degree one in h; (ii) lim Ilh11111f(x+h)-f(X)-Of(X,h)II=O far all xED. 11411-0

Theorem 3.3.3. f(x)

!8

G-dtfferentUlble in the fuuJRly open

set D if arul only if far

eL'e1'Y

xE

D and hEE,f(x+zh) is a Teqular functIOn of z when, Izl
Proof. This follows from the observation { :zf(X+ (Zo+Z)h)}

.=0

=!~~!

{f( (x+zoh) +zh) - f(x+zoh)} Q.E.D.

=of(xo+zoh ,h).

If f (x) is G-differentiable in the finitely open set D and if xED we may define the

n-th variation O"f(x,h) of f(x) with increment h as (3.3.3) The Taylor development of the function f(x+zh) is ~

/(x+zh) =

~

[dd°.f(x+ah) ]

0-0

for

Izl
we obtain

a

;:", a=on.

VECTOR-VALUED ANALYSIS

147

=

f(x+zh)=

~-.LO"f(x,h)z"= ~-.LO"f(x,zh) .

• -0

n!

.=0

nl

Hence the following result holds. Theorem 3. 3. 4. Let f(x) be G-differentiable in the finitely open Set D. Then fur xE D and x+ II

U!

the C-stm about x in D, u{' have f(x+h)=

~n\O"f(X'h).

(3.3.4)

Theorem 3.3.5. Let f(x) be G-differentiable in the finitely open set D and p(x,h»1. If

IIf

(x) II~M, in any C-star C· (a) about x=a, then II000f(a,h) II~Mn! fur a+hEC' (a).

d· ] 0' f(x,h) = [ -dJ(x+zh)

(3. 3. 5)

f

n! f(x+zh) (3.3.6) =-2· +1 dz, z .-0 1ft (' z" where C is any circle C(O,pI ) with p'
(3. 3. 5) holds. Theorem 3.3.6. Let f(x) be G-differentzabIR in the open domain D. If f(x) =0 tlrroghotd some sphere, then f(x)=() in D.

Proof. Suppose that f(x)=() throughout the sphere S.: IIx-all
that O'j(a,x-a)

=()

for all x. It follows from Theorem 3.3.3 that f(x)

=()

in the

largest sphere S with center at x=a which is contained in D. Now if b is any point in D, we may join the points a and b by piecewise continuous line segments and hence by a finite chain of open spheres in D, So = S , S 1 ,

••• ,

S., such

that S. contains the center of SHI. Since f(x)=() in So, the preceding argument shows that f(x)==() in SI' and hence by induction, f(b)=(). Thus f(x)=() in D.

Q.E.D.

Remark. The content in this section can be extended to a locally convex space by the

method in next section.

CHAPTER

148

3

3.2. VECTOR-VALUED REGULAR FUNCTIONS IN LOCALLY CONVEX SPACES

Let E be a complete Hausdorff locally convex space on the real or complex domain K, and P be the sufficient directed set of seminorms which generates the topology of E.

Definition 3. 3. 6. we say that a sequence {x.} of elements in E is convergent strongly to Xo E E, if for any

pE P,

lim p(x.-xo) =

O.

Defintion 3. 3. 7. Let E be a complete Hausdorff locally convex space and D be a domain in the complex plane C. Suppoes fez) is a function from D to E. (i) fez) is called a vector-valued regular function on d if for any rpEE' ,rp(f(z»)

is a numerical regular function, where E' is the dual space of E. (ii)If for any z in D and any p in P there exists AE E such that (f(z+a) ' Zunp ..-0

- fez)

a

A)

exists, then fez) is called strongly differentiable at z, and denoted by f' (z). fez) is called strongly differentiable on D if fez) is strongly differentiable at each point z in D. Theorem 3.3. 7. Let E be a WcaI1y convex complRte linRar topological space. fez) is strongly differentiable in D if and only if fez) is a vectm-vab.led requJar function.

Proof. The necessity of the Theorem is obvious, we only prove the sufficiency of the theo-

rem. Let S be a compact subset in D. For any

rpE E'

there exists a constant m(rp,S) such

that for z,z+a,z+p in S

Ia 1 p{ ! (rp(J(z+a)-f(z»- ~ (rp(J(z+P)- f(z»} I<,m(rp,S). Put

f(z,a'fJ)=~p{l...(f(Z+d)-f(z»aa

pI (f(z+P)-j(z»}.Then

J= {J(z,a,p) : for any z,z+a,z+PE S} is a weakly bounded set with elements in E. Since E is a complete locally convex space, J is a bounded set in E. Without loss of generality, we may assume U is a absorbing set, so there exists a in

q;'

such that J Cal!, i. e.

1 1 -a(J(z+a) - f(z» -p(f(z+ P) - f(z» E (a- P)aU.

VECTOR-VALUED ANALYSIS

149

Since U is a balanced absorbing set, (a- P)aUCu for sufficiently small a and p. By the completeness of E we get that the strong derivative fl (z) exists. It follows that .l(f(z+a)-f(z))-f' (z)EU as

a

p-O.

Q.E.D.

We can extend theories about vector-valued regular functions in Banach spaces to locally convex complete spaces. For example the theorems of Sec. 3. 4 hold in locally convex complete spaces. In order to better explain these results, in the following we extend the theories of Sec. 3. 5 to complete locally convex spaces. Theorem 3. 3. 8. Suppose flint D is a simply connected domain, that E is a complete seminorm space, and that the set {f(z)} of vector-valued regular functinns from D to E is uniformly bouruled an some compact set SinD. Then {f} is equicontinuous

0/1

S.

Proof. For any Zo E D and pEP by the uniformly bounded condition we conclude that

there exist M and r such that p(f(z))<"M

as

(3.3.7)

zES(zo,2r).

From the vector-valued Cauchy integral formula we obtain

~ _1 p(f I ("'))-P(2 . 1ft

J. C(%o,2r)

dt (tf(t) - z )2 ),

-

(3. 3. 8) 2M

where zES(zo,r). (3.3.7) and (3.3.8) show that p(fl (z))<,,-. Since S is a comr

pact set, p(fl (z) )<"M " as

zE s,

where M 1 is a constant. Applying the vector-valued Cauchy theorem we have f(z.)-f(z,)=

J" f'

(z)dz

for Z"Z2 in S.

'I

It follows that for any p in P p(f(z2)-f(ZI))<"M 1 I z 2- z ll,

thus {f} is equicontinuous on S.

Q.E.D.

Theorem 3. 3. 9. If E is a complete Wcally convex space, then Thearem 3. 1. 19 holds in E.

CHAPTER

150

3

Proof. Suppose that S is any compact in D and that {f} is normal on D. For any e>O, any {z.} CS and any {f.}C {f} by the hypothesis there exist !Y, fo(z) and {f.,}C {f.} such that p(f., (z) - fo(z) )<e

as

zE 8 and k>S.

(3.3.9)

It follows from a result (see Sec. 7. 2 of [34] ,Theorem 5) that there exist positive num-

bers al , •••

,a. and seminorm PI' ••• ,p. E P such that Itp(f., (z) -

.

fo(z))

I~ L; (JiPi(f. (z) -

(3.3.10)

fo(z))

i=1

z in S. so {tp(f., (z))} uniformly converge to tp(fo(z)). Thus tp(!o(z)) is a numerical

regular function. It follows that fo(z) is a vector-valued regular function. Without loss of generality we may assume z', - Zo as k-oo. Hence for any pEP, where {z.,}C{z.}CS

Since {f., (z)} is uniformly convergent, there exists 1'2 such that I p(f.,(z)-fo(z))<2"e

as zES and k>:'Y2.

(3.3.12)

(3.3.11) and (3.3.12) give p(J., (z.) - fo(zo) )~p(f., (z.) - !o(z.))

1

+ p(Jo(z.,) -

fo(zo))

1

~2"e+2"e=e.

Therefore {f. (z.)} has a convergent subsequence, 1. e. the range of {f} in any compact set SCD is sequentially compact. If the range of {f} in any compact set SCD is sequentially compact, then {f} is

normal on D. The proof is same as that of Theorem 3. 1. 19.

Q. E. D.

BIBLIOGRAPHY

1. Ahlfors, L. V. , Complex Analysis, 3rd Ed. , McGraw- Hill Book Co. , New York, 1979. 2. Caratheodory, C. , Funktionentheorie, Vols. I and

n. Verlag Birkhauser, BaseI,1950.

Translated by Steinhardt, F. , Theory of Functions, Chelsea Publishing Company, New York, 1954. 3. Conway, J. B., Functwns of One Complex Variable, 2nd ed. , Springer- Verlag, New York, 1978.

4. Curtain, R. F. and Pritchard, A. J. , Functional Analysis in Modern AppZi£d Mathenwtics, Vol. 132 in Mathematics in Science and Enginearing, Academic Press Inc. ,London,

1977. 5. Diestel, J. , Sequences and Senes

In

Banach Spaces, Springer-Verlag, New York, 1984.

6. Diestel, J. and Uhi, J. J., Vector Measures, the Amer. Math. Soc. , Providence,

1977. 7. Dineen, D. , Complex Analysis in UxalJj,y Convex Spaces, North-Holland Publishing Company Amsterdam, New York, 198!. 8. Duren, P. L. , Theory of H'-space, Academic Press, New York, 1970. 9. Gelfand, I. , Abstrakte Funktionen und lineare Operatoren,

MaTeM

c6. 4 (1938), 235-

284. 10. Griffel, D. H. , AppZi£d Functional Analysis, Ellis Horwood Limited, 198!. 1l. Hille, E. and Phillips, R. S. , Functional Analysis and Semi-Groups, the Amer. Math

Soc. Colloq. Publ. , Amer. Math. Soc. , Providence, R!. , 1957. 12. Hoffman, K. ,Banach Spaces of Analytic Functions, Prentice- Hall, Englewood Cliffs, N. J. , 1962.

152

BillLIOGRAPHY

13. Hu, C. G. , A necessary and sufficient condition of the normal family of vector-valued holomorphic functions, Acta Sci. Natur. Univ. Nanl«Jiensis, 1(1981) ,27-30. 14. Hu, C. G. , Vector-valued integrals of Cauchy type and their applications to vectorvalued singular integral equations, Acta Sci••T\Tatur Univ. Nanl«JlRnSis, 1 ( 1982), 4858. 15. Hu, C. G. , Boundary properties of a class of vector-valued analytic function in the unit disc, Acta Sci. Natur. Unw. Nanl«Jiensis, 2(1984), 109-116 16. Hu, C. G. , Advance of vector-valued analytic functions and its applications to vector-val ued singular integral equations, AppliEd Mathematics and Calculated Mathematics, 3 (1984), 95-110. 17. Hu, C. G. , Complex Analysis, Nankai Univ. Publ. House, Tianjin, 1985. 18. Hu, C. G. , Vector-Valued Analysis and Their Applications, Nankai Univ. Publ House, Tianjin, 1985. 19. Hu, C. G. , The disturbance of vector-valued doubly-periodic Riemann boundary value problems, J. Math. Anal. Appl. , 131(1988), 373-391. 20. Hu, C. G. , Vector-valued boundary value problems of a class, in "Proceedings of mternatinnal Conference on /nteqral Equations and Buundary Value Problems" (ed.

fhR

by G. Wen

and Z. Zhao), World Sci. , Singapore, New Jersey, London, Hong Kong, 1991, 57-64. 21. Hu,C. G. and Beg,I. , Cauchy-StieJtjes integral in locally convex spaces, Complex Varlr ables, 15(1990), 233-239.

22. Li, G. P. , Guo, Y. Z. and Chen, Y. T. , Automorphic Functions and Minkowski Functions, Science Press, Peking, 1979.

23. Lindenstrauss, J. and Tzafriri, L. , Classical Banach Spaces /, Springer-Verlag ,Berlin , Heidelberg, New York, 1977. 24. Lu, J. K. , Immediate methods of the finding solutions of integral equations, Acta Sci. Natur. Univ. Wuhanensis, 1 (1975), 12-27.

25. Lu, J. K. , Doubly-periodic Riemann boundary value problems, Acta Sci. Natur. Univ. Wuhanensis, 3(1979), 1-10.

26. Lu, J. K. , Buundary Value Problems of the Complex Analysis, the Science and Technology Pub!. House, Shanghai, 1987. 27. Markushevich, A. I. , Theary of Functions of A Complex Variable, Chelsea Publishing Company, New York, 1985.

BiBLIOGRAPHY

153

28. Muskhelishvili, N. I. , Sinqular /niRqral Equatinns, 3rd. ed. , Nauka, Moscow(in Russian), 1968. 29. Natason, N. P. , Thearyof Real Variable, The National Technological Theory Press of USSR, Moscow (in Russian), 1957. 30. Riesz, F and Nagy, B. Sz. , FunctW1U1l Analysis, lRcons D' analyse FonctinnRlle, Translated from the 2nd French ed. by Leo F. Boron, Budapest, 1956. 31. Schaeffer, H. H. , TopoWgical Vector Spaces, Graduate Texts in Math 3, Springer-Verlag, New York, Heidelberg, Berlin, 1971. 32. Segal, Sanford L. , NirIR mtroductinns

!II

Camplex Analysis, North-Holland Publishing

Company-Amsterdam, New York, Oxford, 1981. 33. Wermer, J. , Banach Algebras arul Several Camplex Variables, Springer- Verlag, New York, Heidelberg, Berlin, 1976. 31. Wilansky,A. , Modern Methods in Topological Vector Spaces, McGraw-Hill, New York, 1978.

INDEX OF SYMBOLS

A(r,f)

1

b(r,f)

1

E

94

43

E'

91

Eo

JOO

S-(z)

116

fJ-~

q?

C(f) C6

J 46

94 7 39

F

II j II, II j II ~ D

3

Do

4

26 87 87

Dj(zo)

34

G(z)

43

D(p,r)

21

Go

43

dQ

4

d(WI,W2)

27

J(WI,W2)

28 28

HI

90

H'

68

4 42

H~

68

h'

145

h'

68 68

J(z,=)

l::,.u

00

o(x,h)

155

INDEX OF SYMBOLS jzl

[{P

[{(o,J(O) )

[{(JCO) )

69

0.( {f} ,zo)

38

0.( {f} ,zo,o)

38

P(z)

25

92 8 51

p( Ico

I>

9'(z)

U(U)

86

LaCI)

12

log+x

2

log-x

80

q(r,f)

gz

15 55

1

Rt

94 64

R.

64

M(r,J)

68

res f(a)

Mp(r,j)

68

rt

106 64

M.~(r,f)

76

r.

63

S(r,f)

1 39 61

m(r,j)

1

ll/z(r,j)

1 S6 U

S

79

SeT)

13

XCr,a)

1

X(r,oo)

1

S(7',J)

1

S(r,l/(j-a)

T(r,f)

1

U

137

1

V

143

1

Var(C)

103

n(r ,J)

1

n(r,oo)

1

nCr ,a) nCr, l/(j-a»

INDEX OF SYMBOLS

156 Var {
95 123

Q(f(O) ,0)

42

QI(6,O)

42

113 100

Q2

51 52

W,

27

W,.

27

Q.... (ko,k l

II II

II" II v

, .. •

,k.,O, R)

123 143

INDEX

abnormal point

38

absolute constant absorbing set Ahlfors

Cauchy integral formula

48

Cauchy principal value compactness

148

52

arithmetic mean

8

conjugate Poisson kernel

112

convergent uniformly converges sphericmaly Banach space Bieberbach

101 92

52

Bloch

52

64

convex function

2

145

derivative

52

weak

Bloch constant

49

Bochner integrable Bochner integral boundary value

102

strong

Bloch-Valiron-Landau method

bounded

convex constant C-star

Blaschke product

99 99

47

102

differentiable weakly strongly

69

domain

102 102

3

146

bounded variation

90 Egorov' s theorem entire function

Caratheodory,Bloch

91

continuous convex function

89

Banach algebra

117

107

complex plane

approximate identity

104

52

equicontinuous

47

86

41 25

3

INDEX

158 spherically

32

Julia direction

55

Julia exceptional function Fabry

58

56

Fatou' s lemma

81

F -differentiable

146

finitely open

Landau Laplacian

145

first variation

47 4

Laurent expansion

16, 106

145

Frechet differential

Lebesgue' s bounded convergence theorem

146

83 31

limit point Gap theorem

linear space

41

G-differentiability geometric mean

Liouville's theorem

145

locally convex space

4

Hardy's convexity theorem

47 , ] 06

Littlewood-Collingwood-Nevanlinna

89

Green's theorem

68

LI-norm

71

U-norm

92

21

148

76

harmonic function4 Hartogs theorem

120

Helly' s selection principle

maximum principle 69

Herglotz representation theorem homogeneous

146

Hurwitz-Polya

infinite order inner function

meromorphic 70

106

1

monotonic increasing multiplicity

6

25

56

Natason

56

69

Nevanlinna' s second fundamental theo-

89

isolated essential singularity

rem

57

21

non tangential limit normal

72

27,109

Jensen's formula

80

factorization theorem

Jensen's theorem

81

family

26

89

INDEX

159

operator

4

"sectorial" limit

5

order

semi-norm

Ostrowski's theorem outer function

40

sequence

89

overconvergent

73

148 26

of regular functions

40

singular inner function

31 89

56

singularities space Parseval formula

76

Picard theorem point

47

finite measure

98

topological spherical

31

J-

39

derivative

0-

39

distance

u>-

7

equicontinuity

formula

Poisson kernel

80

26 32

64

steregraphic projection

107

Polya

34

starlike constant

69

Poisson-Stieltjes integral pole

98

98

a-

Poissons

93

complete metric

69

strong bounded variation

54.56

continuous

p-valent function

64

regular

71

subdomain

26

closed

26

subordination

78

3 Theorem

106

Riemann sphere rotation

68

3

regular function residue

94

97

subharmonic function radial limit

28

12

12

(Ascoli)

32

(Bloch)

47

(Cauchy) (Julia) Schauder basis

115

Schottky theorem Schwarz lemma

42 78

(Landau)

103 54 51,65

(Littlewood)

78

(Littlewood- Collingwood-

]60

INDEX

Nevanlinna)

21

(Marty)

58

(Nevanlinna)

15

disturbance problems

133

value problem

123

elliptic function 113

81

(Schottky) (Valiron)

79

52

(F. Riesz)

117

doubly- periodic Riemann boundary

(F. and R. Nevanlinna) (Picard)

Cauchy type integral

function

42

91

Holder condition

65

integration

100

96

meromorphic function

transformation of rotation

]2

Plemelj formula

1 L8

regular function

102

I. ] 3

singular integral equation Vivanti-Dien!.e

56

105

uniquene!.S theorem univalent disk

19

function

weak bounded

18

universal constant

variation

61

!.et

91

118

WeierstraS!.' s elliptic function Valiron-Milloux vector-valued

57

function

116

theorem

51

55

119