SOME

PROBLEMS V.

A.

ON N O N O V E R L A P P I N G

DOMAINS UDC 517.54

Andreev

INT RODU CTION Let ~ be the class of the p a i r s of functions if(z), F(~)], w h e r e f ( z ) is such that f(0) = 0 is a r e g u l a r univalent function in the disk E: lzl < 1 and F(~) is such that F(~) = ~ is a m e r o m o r p h i c univalent function in the domain K: t~l > 1, mapping E and K, respectively, on nonoverlapping domains ]3 and D of the plane Cw, We will say t h a t f ( z ) iF(i)] belongs to the c l a s s S" (Z') if there exists a function F(~) in Z~ such that if(z), F(~)] ~ .

in S)*

The e x t r e m a l p r o b l e m s on ~ , S', and Z" are related to a large complex of questions in problems on nonoverlapping domains. A survey of the l i t e r a t u r e on these p r o b l e m s up to the p r e s e n t time, of course incomplete, is given in [1] (Appendix). We will note only three r e s u l t s . N. A. Lebedev [2] and G. V. Ulina [3] have considered, respectively, the p r o b l e m s of finding domains of values of the functionalsf(z0)/F(~0) (z 0 ~ E and ~0 ~ K are fixed points) and [f"(0)/f'(0)l+ i ] ( ' ( 0 ) / F ' ( ~ ) t o n ~o With the first of these is closely connected a result of [4] and with the second is closely connected T h e o r e m 7.3 of [5]. K~/hnau introduced the c l a s s S" in [6] and solved the e x t r e m a l p r o b l e m for If'(0)l for a fixed Jr(z)Ion it and by the same token actually found the domain of values of the functional If(z0)/F'(~)I+ i ! f ' (0)/F ~ (~)! on ~ . In the p r e s e n t article we will consider two types of functionals on ~ depending on the quantities f'(z0) and FV(~0), andf'(z0), f"(z0), and F ' ( ~ ) (z 0 ~ E, ~0 ~ K ) and their conjugates. General r e s u l t s in the p r o b l e m on finding the domains of values of these functionals, formulated in T h e o r e m s 1 and 2, will be obtained. In many concrete c a s e s of this, it is sufficient for the complete eIucidation of the e x t r e m a l p r o b l e m s . In p a r t i cular, the solutions of the above-mentional p r o b l e m s in [3, 5] can be easily obtained f r o m T h e o r e m 2 (there are m i s p r i n t s in the f o r m u l a s of [3]). As a c o r o l l a r y , results for the c l a s s S" are obtained. The investigation is c a r r i e d out by the variational method by using the variational formulas of [2]. w

FORMULATION

OF

THE

PROBLEMS

For the solution of v a r i o u s p r o b l e m s on ~ it turns out n e c e s s a r y to consider its closure

~l.

Let (fn, Fn), n = 1, 2. . . . . be an a r b i t r a r y sequence of p a i r s of ~], and (Bn, Dn) be the corresponding sequence of p a i r s of domains in CwLet us choose f r o m (Bn} and {Dn}, on the b a s i s of the l e m m a of [7, p. 130], all possible subsequences {Bnk} and {Dnk#, respectively, converging to degenerate as well as nondegenerate kernels with r e s p e c t to the points w = 0 and w = c o , and let us then f o r m the subsequences {Bnkp, Dnkp) [we will denote them by (Bin, Dm)]. If the kernels of {Bm} and {Dm} are both nondegenerate, then by- C a r e t h g o d o r y ' s t h e o r e m {fro, Fro) has a limit pair (f, F) in ~ .

* The set of all functions f(z) = z A function F(~)E E0 if F(z-1)-t~S.

+

e2Z2 +

o.

,

which are holomorphic and univalent in E is called the c l a s s S.

T r a n s l a t e d f r o m Sibirskii Matematicheskii Zhurnai, VoI. 17, No. 3, pp.483-498, May-June, 1976. Original article submitted July 15, 1974. mThis material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 1 7th Street, N e w York, N.Y. 10011. N o part ] o f this publication m a y be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic mechanical, photocopying, icrofilming, recording or otherwise w i t h o u t written permission o f t]le publisher. A cop3, o f this article is available from tl~e publisher for $7.50.

I 373

If the kernel of {Bin} ({Din}) is Cw(C-w/{0}), then {fm} ({Fm}) converges uniformly in E\{0} (K \{~}) to (to 0), and in this case we write (fm, Fro) - - (% :~ Fm) -- (0, 0)). Considering also the other c a s e s of degenerate kernels, we obtain the limit p a i r s (o% oo), (/(z), o~), (0, co), (0, F(~)), (0, 0),

(I)

where f(z) and F(z-l) -i belong to the class M of all those functions ~o(z) which are holomorphic and univalent in E and for which q~(0) = O.

As an example let us o b s e r v e that for an a r b i t r a r y pair (f, F)~ ~ t h e anF) is (0, 0) o r (~, ~) according as {an} converges to 0 or r

limit pair of the sequence (anf,

Denoting the family of the p a i r s (1) by ~t', we set ~ =~t.) !0t'. Let I

= J ( z 1. . . . .

Zn)

be an analytic function of its v a r i a b l e s in a sufficiently large domain and the func-

tional 1 = ] ( .... um Izi~,..., Ukl, U~I, ...),

u~j=f')(zj), Uk~-F
is

Let L j ( L ' ) denote that part of the boundary 0Gj which is introduced by the p a i r s of ~/ (!l)l'). We will call e v e r y point I0~L J a nonsingular boundary point in the usual sense [8]. Let Ie be an ext e r i o r point of the domain of values G j corresponding to I 0, x = e-iT with 3' = arg (I0-Ie), if(z), F(~)] be the pair of N introducing I 0 and [f, Cz), F,(~)]E ~ b e the variational pair /.(z) ~--~](z)-[-tP(z) -[-o(t), F (~) ~ F ( ~ ) -[-tQ(~) +o(t) (t > 0 is sufficiently small).

With the help of this variational formula we obtain the condition

ReI in the usual m a n n e r .

Here

(*)

+

p~j=pc,>(zj), Ohz=Q r

0 Plj = x . Ou~i ~

(2) o~-~

Definition. If t h e r e exist I e and if(z), F(~)] such that all the quantities Pit and qk/ are not simultaneously zero, then we will call I 0 a boundary point of the f i r s t kind. E v e r y w h e r e in the sequel e v e r y pair introducing such a point will be called a boundary pair. Let us consider on N the p r o b l e m s of finding the boundary points of the f i r s t kind for functionals of the following two f o r m s :

I~--.r

]'(zo---), U(~o), ~"(go)),

T=](/'(~,o), ?'(~),/"(~o), f'(=o), F'(~), F'(~) ),

~)

where z0~E and ~0~K. Since whenever the pair if(z), F(~)] belongs to the class ~ , the pair [f(eiaz), F(eifi~)] with a r b i t r a r y r e a l n u m b e r s a and/3 and the pair [f0(z), F0(~)] with

1o(~) =/((,o-Z)/(t -~0~) ) -1(~o), F0(~) ----F(~) --1(z0), also belong to the class

~ , we get the following equivalent initial p r o b l e m s for functionals: 1=sC/'(r), ]'(r), P'(p), F'(p)), r = I zol, p = I~01, X=J(I'(0), 1"(6), I"(0), I"(0), F ' ( ~ ) , F'(~o) )

(in general, with other functions J).

PROBLEM

(BO)

A

For the boundary p a i r if(z), F(~)] introducing the point I 0 the condition (2) a s s u m e s the f o r m

tle{pP'(r) +qQ' (9) } > 0 ,

374

(A)

We will solve them. w

1.

(3)

(4)

w h e r e p =x(OJ/Of') + x(OJ/Of'),q=x(OJ/OF ') + x ( 0 J / O F ' ) , a n d x is d e t e r m i n e d , a s above, b y a p o i n t I e in the e x t e r i o r of G j c o r r e s p o n d i n g to I 0. H e r e a n d e v e r y w h e r e in t h i s s e c t i o n f ' = f ' ( r ) a n d F ' = F ' ( p ) . A n a l o g o u s l y , f o r b r e v i t y , we w i l l w r i t e f, f", F, a n d F " . L e t us t a k e c o n c r e t e v a r i a t i o n a l f o r m u l a s [2] (t > 0 is s u f f i c i e n t l y s m a l l ) . 1) f ( z ) = f ( z ) + t A f ( z ) / ( f ( z ) - - w ' ) , F,(~) = F ( ~ ) + t A F ( r and D (A i s an a r b i t r a r y c o m p l e x n u m b e r ) . 2)

3)

t __[A/(z') ] t # , (~) {A/(z') ~,51;, ) 9 ~ - ~, [~,/,,<)j

f . (z) = I (z) + t : ( ~Ai ) _ :(~) (,)

t F ( AP(~) ~)--!(~') '

if

if w ' i s an e x t e r i o r p o i n t of both B -

1 z__, .} -~

+o(t),

F,(~)---F(~)~

z'~E.

!,(z)=/(z)+t

[ AF(~')~.F,~(: ')1 ~____L_l ~_~%t+o(t),

! ( z )A!_ F(z)( U ) if

'

F . (~) = F (~) + t

F (~)-- F (~')

(~,~F, ~ (~,) " ~ --

~"

~, ~ K .

W i t h the h e l p of f o r m u l a s 1) we at o n c e s e e t h a t t h e r e e x i s t p o i n t s which a r e e x t e r i o r p o i n t s of b o t h the d o m a i n s B and D. T h e f o r m u l a s 2) a n d 3) a n d c o n d i t i o n (4) l e a d to the f o l l o w i n g e q u a t i o n s f o r b o u n d a r y p a i r s by v i r t u e of the a r b i t r a r i n e s s of z ' , ~ ' , a n d a r g A:

P!' (/(z)--/)--*+qF'(/(z)--FP = z/"(z-------(-[(z--rp -t- (l__rz).~ n-

l P]' (F (~)t --/)2 +qFf(F(~)_t,,).,:-= ~

'1

z--r

-}-

{ (~_p)2 ~9qF" -~ (~Z~)',. ~pqF" -Jr pV(F'@pF")

t--rz

j

q (F' @ p],'")t

T a k i n g the l i m i t in t h e s e e q u a t i o n s a s ~ a p p r o a c h e s :~ a n d z a p p r o a c h e s 0, we g e t t h e r e l a t i o n s

p]'-l-qF'-=O, pfl'~---r2pf"+ 2rpfl. S i n c e the p a i r ~f(ei~tz), F(eifit~)] b e l o n g s to N f o r a r b i t r a r y r e a l n u m b e r s t, a , and/3, f r o m the a b o v e e q u a t i o n s we s e e t h a t the q u a n t i t i e s p f ' , pf", q F ~, a n d q F " a r e r e a l and h = 1 + rf"/f' = (1 + r 2 ) / ( 1 - - r 2 ) . On c o n s i d e r i n g the p a i r s [f(zte), F ( ~ t - 5 ) ] f o r tE (0, 1], e, 6 > 0, we c o n c l u d e that

p]'O, H=t+pF"/F'~O. B y v i r t u e of t h e s e f a c t s t h e e q u a t i o n s f o r b o u n d a r y p a i r s can b e f i n a l l y w r i t t e n in t h e f o r m

G[l(z)]fl2(z)=q(z), G[F(~)]F (~)--Q~(~), where

G(w) = 2 ( f - - F ) (w--wo)/((w--]) (w--F))2, wo= (]+F)/2,

(5)

q(z) =-- (t--r~)2/((z--r) ( t - - r z ) ) 2, Qs(~) =Q~(s-t-s - t -

r~-~-,)l((r~-o) (ti-p-')) ~, O.=(p+p-')~., x=t+~z/, ~ =(o~-,l)/(p2+l),

s-t-s-'=o+p-'-(p-p-')~l'~..

B y v i r t u e of the w e l l - l m o w n i n e q u a l i t y of [1], d e s c r i b i n g the d o m a i n of v a l u e s of the f u n e t i o n a ! • = ~ F " ( ~ ) / F ' ( ~ ) on t h e c l a s s ~, we s e e t h a t 1 _< s < (p+p-t)/2. F r o n t the a n a l y t i c t h e o r y of d i f f e r e n t i a l e q u a t i o n s it f o l l o w s t h a t the f u n c t i o n s f ( z ) a n d F ( ~ - ) d e f i n e d b y E q s . (5) a r e r e g u l a r in E a n d ~\{~o}, r e s p e c t i v e l y , e x c e p t a f i n i t e n u m b e r of a l g e b r a i c s i n g u l a r p o i n t s l y i n g on the unit c i r c l e . tn the u s u a l m a n n e r ( s e e , e . g . ,

[9, p. 36] we e s t a b l i s h t h a t only the f o l l o w i n g t h r e e c a s e s a r e p o s s i b l e .

a) w 0 is an i n t e r i o r p o i n t of the d o m a i n D. r e g u l a r on the unit c i r c l e .

Then s >1,

F ( s ) = w 0, and the f u n c t i o n s f ( z ) and F(~) a r e

b) T h e b o u n d a r y c o n t i n u u m d e f i n i n g the d o m a i n s B a n d D i s a c l o s e d a n a l y t i c J o r d a n c u r v e e x c e p t a b r e a k at the p o i n t w 0 with the i n t e r i o r angle e q u a l to 2~r/3. In t h i s c a s e s = 1 a n d w 0 = F ( s ) . c) T o the c u r v e c o r r e s p o n d i n g to the c a s e b) at the p o i n t w 0 is j o i n e d a s l i t a l o n g an a n a l y t i c J o r d a n a r c in t h e d o m a i n D. T h e a r c s c o n v e r g i n g at w 0 f o r m e q u a l a n g l e s and s = t and w = F ( s ) a r e the e n d s of the slit.

375

2. F u r t h e r investigation will be c a r r i e d out by the m e t h o d s s i m i l a r to t h o s e of [5], which allow us to o m i t s e v e r a l d e t a i l s . Let w 1 = ( F - - f ) / 2 ,

=f

and l be a r a y with the initial point at w

i "mo

and p a s s i n g t h r o u g h w = F.

The function

V~---:~o

(6)

wa

f o r the a p p r o p r i a t e c h o i c e of the b r a n c h of the r a d i c a l , m a p s G o =Cw~l on the following domain of the plane CW, W = U + i V : {W:U~<0,

O
F u r t h e r let w(z) = l o g [ ( r - - z ) / ( 1 - - r z ) ] . H e r e and in what follows, if not s t a t e d s p e c i f i c a l l y to the c o n t r a r y , only the p r i n c i p a l value of a l o g a r i t h m will be taken. The function W = w (z) m a p s the disk E with a slit along the s e g m e n t [r, 1] on the h a l f - s t r i p { W : U < 0, --Tr < V < 7r}, w h e r e co(0) = l n r . L e t us c o n s i d e r the m a p p i n g W = P.,(~)= i

i + ~ - l - - ~ - -1s /-2~,), .

(7)

s

I f s = 1 , then QI(~) = 2i ~ t

j'(~

~--1 . d~ _ p ) ( ~ _ p -~) ~ "

i

L e t K 0 denote the d o m a i n K with a slit along the r e a l axis to the right of ~ = 1 and let us c h o o s e in K 0 t h o s e b r a n c h e s of the r a d i c a l s (~ + ~ - l _ s - - s - i ) 1/2 and ~i/2 which a s s u m e p o s i t i v e v a l u e s at the point~ =p on the u p p e r edge of the slit. T h e n the m a p p i n g (7) for s = 1 t r a n s f o r m s the d o m a i n K 0 into the v e r t i c a l s t r i p

{W: 0 1 it t r a n s f o r m s K 0 into the domain { W : - - [ I o < U < ~ 0 , 0 < V < 2 ~ [ I } U {W : 0 < U < a } . Here TI~

2T

Ilk

f 't / - ~x ~ -s_ l

8(T--l,k)

.

d=

"l--~x:'

1

(8)

1

H:

2T

~ 8(x,k)

8 (~-1, k~ ~

,

ax,

0

6(x, k)=]/(l--k2x2)/(l--x2), k=2~s'7(t+s),

~=2]/~(l+p).

L e t us o b s e r v e that in the c a s e s = 1 the s e g m e n t of the i m a g i n a r y axis f r o m 0 to 2i In L, w h e r e L = ((1/p-+~)/(~p--l))L

(9)

is the i m a g e of the unit c i r c l e . 3. All the m a p p i n g s c o n s i d e r e d above a r e c o n f o r m a l and univalent in t h e i r r e s p e c t i v e d o m a i n s . can now e a s i l y c o n s t r u c t b o u n d a r y p a i r s with t h e i r help.

We

In the c a s e s = 1 the r e l a t i o n s

09(w)--~i~---o)(z), @(w)--~i.-.~-Ql(~)--ilnL

(10)

f o r v -< l n L implicitly d e t e r m i n e the p a i r of functions [fi(z), Ft(~)] m a p p i n g E and K, r e s p e c t i v e l y , on the i n t e r i o r B 1 and the e x t e r i o r D 1 of the c u r v e F 1 which is the p r e i m a g e of the i m a g i n a r y axis f r o m 0 to 2i In L under the m a p p i n g (6). M o r e o v e r , in the c a s e v < l n L t h e r e is a slit in the domain D 1 along the r a y l f r o m W =w0to W =Y. T h i s p a i r b e l o n g s to the c l a s s

376

~l only if the quantity ~t = f i / F l ,

0 < ~l < 1, s a t i s f i e s the r e l a t i o n

y(•

r,

w h e r e ~ l = ~(~l),

•

=i/(1+~)/(1-~), y(z)=n~[(•215

1)} +2,~,.c tg 7..

It f o l l o w s f r o m (9) t h a t the c o n d i t i o n ~ <_ In L is e q u i v a l e n t to t h e i n e q u a l i t y p _< P0 = c ~

(1I) 9

If s > 1, t h e n f o r [[ = ~ t h e e q u a t i o n s (1)( w ) - - a i ~ o ) ( z ) - 1-I0, (l)(w)--~i~(_L(~)--itI

(12)

i m p l i c i t l y d e t e r m i n e the f u n c t i o n s w = f s ( Z ) a n d w = F s ( ~ ) m a p p i n g E a n d K on the d o m a i n s B s a n d Ds w i t h c o m m o n boundary along an analytic curve F s which separates F i f r o m the point w =f. The normalization condition fs(O) = 0 is fulfilled only if U (•

w i t h x s = z (~s), ~s = f s / F s ,

= I n r--Eo

0 < ~s < 1 i n the n o t a t i o n of (10).

It is e a s i l y s h o w n t h a t t h e c o n d i t i o n II =

v,

1

," ti C~, ~)

~ d x

w r i t t e n in the f o r m -~

0

b y v i r t u e of {8), g i v e s p r e c i s e l y o n e v a l u e of k in (0, 1] f o r e v e r y p >-P0. 4. Let us now find the derivatives f'(r) and F'(p) for the pairs of functions constructed above. The following resolutions are considered respecti~cely in the left half-plane of the plane C w determined by the direction of the ray l and in upper half-neighborhoods of the points z = r and ~ = p :

r (w) =log[ (z~,-/)/4~v~] +~/2+O(z,~--/), q) (w) - ~ i log [ (w--F)/4wl] +a+ai/2+O (w--F), (o (z) -~log (z--r) - - l n ( l - - r 2) --~i+O ( z - - r ) , where

m = l / 0 , ~ i r ' ( U - L ( P ) d~, s~>f, T

,(~)~"

(i4)

(~+~-I--s--s-t) '/'f (~--p-1), ~'C2,--=2k/T6(~-', k),

=

a n d O @ ) , O(0) = 0, d e n o t e s a f u n c t i o n r e g u l a r

in a n e i g h b o r h o o d o f the p o i n t ~ = 0.

S u b s t i t u t i n g t h e s e r e s o l u t i o n s in (10) a n d (12) a n d t h e n t a k i n g l i m i t a s z a n d ~ a p p r o a c h to r a n d p, r e s p e c t i v e l y , we g e t .i'j Tl

( p - i) L -

-

1 - - r2

eXl) ( - - -~ - - I I 1 ) ,

if

P~P0;

and /~ =- ~ e x l ) Further,

( - II o - - II,),

if

P

-~Po"

let

jo, o(~) = . [L(z0(~))-L(z0(0) )], F0 o(r = o [F~(r a b e a n o n z e r o c o m p l e x n u m b e r , a n d t h e f u n c t i o n Z 0 (z) b e d e f i n e d b y t h e e q u a t i o n (Z - r)/(1 - rZ) = e i 0 (z - r ) / ( r r z ) f o r 0 -< 0 < 2~r. L e t u s d e n o t e the f a m i l y of t h e s e p a i r s of f u n c t i o n s b y ~2.,. It is easily seen that the pairs of -~t satisfy Eqs. (5) and introduce

I~II=R~, ~, q = f / F ' ,

all the points of the neighborhood

(15)

where

377

1) L

i(P

exp (-- .n - - II1), p ~ Po

_---ZT_,exp(--II0 1 - - II~),

(16)

P ~ P0"

k

M o r e o v e r , for e v e r y p a i r if(z), F(~)] i n t r o d u c i n g a point of the n e i g h b o r h o o d (15) we have i f ( r ) a n d F ' ( p ) = F'O,a(p) for s u i t a b l e a a n d 0. A f t e r r e d e s i g n a t i o n we c a n a s s u m e that

/'e,~ (r) = ae~'ORr,9, F'o,a (p) =

=frO,

a.

a (r)

(17)

5. Now let if(z), F(~)] be a fixed b o u n d a r y p a i r , ](g~)--~F(1), Xo= {z:z ~t#r 0 ~ t ~ i } , F0--~:~-----t, t ~ l } , E o = E \ To, Ko = KIF0, Bo---~j(Eo), Do~F(Ko). L e t u s c h o o s e t h e f o l l o w i n g r e g u l a r b r a n c h e s of log [(z - r ) / ( 1 - r z ) ] , r e s p e c t i v e l y , in E0, K 0, and G O =Cw\[f(T0) U F(F0)]: the b r a n c h v a n i s h i n g for z = 1; (~ + ~ - l - - s ~- s - i ) l / 2 , p o s i t i v e on the u p p e r edge of the s l i t in K 0 to the r i g h t of s; and v ~ - w 0 e q u a l t o i ~/-wtfor w =f. I n t e g r a t i n g the r e l a t i o n

]/G[](z) ]d](z)

~--~-d(log[ (z--r)

/( l--rz) ] )

with r e s p e c t to 3E f r o m eig~ to ei(~o+27r), we get 2rri on both the l e f t - h a n d and the r i g h t - h a n d s i d e s . queatly,

a

--~s j

Conse-

p + p-t__ 2 cos 0

0

T h i s r e l a t i o n , e q u i v a l e n t to (13), shows that s d e p e n d s only on p for p ->P0. If s = 1, then we a n a l o g o u s l y get p < P0. L e t us f u r t h e r a s s u m e that s > 1 and s e t Oo (w) ~- i log A + l o g A,

A = ( W - w0- ~T,)/(Tw- ~0+~/~), w h e r e we take the r e g u l a r b r a n c h e s of the l o g a r i t h m s in the d o m a i n s of the p l a n e s C A and C A c o r r e s p o n d i n g to G o with the n o r m a l i z a t i o n log 1 = 0. F r o m E q s . (5) we get the r e l a t i o n s (I)0[/(z) ] = l o g [

(z--r)/(t--rz) ] +C,

(18)

cr

r iF (;)J = ,[ VO-5;)d; = C (~), which hold a l s o on the unit c i r c l e . Since p

It-- ~F'

-2-~ Re [p~ (~)1 = Re i l/07~" V--F ~ ,

~ 0

f o r ~ = e i~, it follows that Re{,I, 0 [F(~)]} = C l on I~1=1, w h e r e C 1 d e p e n d s only on p.

T a k i n g the f o r m of the

d o m a i n s into a c c o u n t we see that

Re {r a l s o on Izl = 1.

=c~

C o n s e q u e n t l y , C = C 1 + i c , w h e r e c is, in g e n e r a l , d i f f e r e n t for d i f f e r e n t b o u n d a r y p a i r s .

R e s o l v i n g , a s in P a r a g r a p h 4, the l e f t - h a n d and the r i g h t - h a n d s i d e s of the r e l a t i o n s (18) in a p p r o p r i a t e n e i g h b o r h o o d s of the p o i n t s w =f, w = F, z = r, and ~ = p and then f i n d i n g the v a l u e s of the d e r i v a t i v e s f ' and F ' , we get

fl / F ' = R e% w h e r e the r e a l n u m b e r R d e p e n d s only on r and p. We get an a n a l o g o u s s t a t e m e n t for s = 1 a l s o .

378

Thus,

all the p r e c e d i n g c o n s i d e r a t i o n s p r o v e the f o l l o w i n g t h e o r e m .

T H E O R E M 1. A l l the b o u n d a r y p o i n t s of the f i r s t k i n d of the f u n c t i o n a l (A) a r e i n t r o d u c e d b y p a i r s of the f a m i l y ~ , , f o r which the q u a n t i t i e s f ' (r) a n d F ' ( p ) a r e g i v e n b y the r e l a t i o n s (17), (16), (14), and (8). w

PROBLEM

B

In the f i r s t five p a r a g r a p h s of t h i s s e c t i o n we will r e p r o d u c e the p r e c e d i n g s c h e m e f o r the s o l u t i o n of the p r o b l e m B ~ T h e n o t a t i o n u s e d h e r e is n o t the s a m e a s the one u s e d e a r l i e r ; in p a r t i c u l a r , we will now u s e f ' , f " , and F ' to d e n o t e f ' (0), f " ( 0 ) , and F ' ( ~ ) , r e s p e c t i v e l y . 1. As b e f o r e , I 0 i s a b o u n d a r y p o i n t of the f i r s t k i n d f o r O j i n t r o d u c e d b y the b o u n d a r y p a i r if(z), F(~)t a n d I e is a p o i n t in the e x t e r i o r of G j c o r r e s p o n d i n g to I 0. C o n d i t i o n (2) can b e w r i t t e n in the f o r m Be {p~P'(0) -}-p~P"(O) + q Q ' ( o o ) } >/0, o:

p~ = x ~

, - os

= ~ ~,

os + ~ os p~ = x ~ ~,

oJ + ~: o s q = x ~ ~p-.

0.9)

F r o m f o r m u l a s 2) and 3) a n d c o n d i t i o n (19) we g e t the f o l l o w i n g e q u a t i o n s f o r b o u n d a r y f u n c t i o n s : Pt]' ', p,_ (.i"/ (z)4-2/'"). / (z) ' /" (z)

f(z) [2p:f' + ( p f 2_~2p~J....j z + 2p,~z~], ='V': (z) " "

p,!" po (r',F (~) + 2/") _, e(~) e(~) + - F~ (~) = - - q*" ~2F"*(~" T a k i n g l i m i t in the a b o v e a s ~ a p p r o a c h e s oo, we g e t p~]'-4:-p~fl'-}-q F' = O.

On c o n s i d e r i n g the a u x i l i a r y f o r m u l a s of P a r a g r a p h 1 o f w

it f o l l o w s that

p f -b-2p:]" ~ O , q F ' ~ O .

(20)

T h e n the e q u a t i o n s f o r a b o u n d a r y p a i r a s s u m e the f o r m

c[/(~) ]I'~(~) =~(~-~)(~-~-')/~, G [ F ( ~ ) ] F '2 (~) = 1/~ ~,

(21) (22)

where G (w) = (u'--u'o)/u '3, u ' o = --gfl, g--~. - - 2 p S / q F ' , g (,u+~t -I ) = (Pl[-[-2P2[")/qF'=p~J"/qF'-- i.

(23)

g = t e ~, ~ ~ Te'*

(24)

Let us set

T h e n f r o m (23) and (20) we get e i ( r

o) = 1.

A s b e f o r e , we c o n c l u d e that t h e r e do n o t e x i s t p o i n t s e x t e r i o r to b o t h the d o m a i n s B and D, which can b e of o n l y the f o l l o w i n g t h r e e f o r m s : a) w 0 i s an i n t e r i o r p o i n t of t h e d o m a i n B . 1 and I~t = 1, r e s p e c t i v e l y . curve.

Then g E E , f(p) = w0, a n d f { z ) and F(~) a r e r e g u l a r on t zL =

b) w 0 is an a n g u l a r p o i n t of the c o m m o n b o u n d a r y of the d o m a i n s B and D, which is an a n a l y t i c J o r d a n T h e i n n e r a n g l e at t h e p o i n t w = w 0 is e q u a l to 47r/3, [p} = 1, andf(p.) = w 0.

e) To the curve of the case b) is joined at the point w 0 a slit in the interior of the domain B along an analytic Jordan arc. All the angles with vertices at w 0 are equal to 2~/3, tp[ = I, andf(p) is the end of the slit.

2) L e t us at f i r s t c o n s i d e r the c a s e Pl ~ 0, P2 m 0, q ~ 0. L e t u s d r a w in the p l a n e C w a s l i t along t h e r a y I f r o m w = 0 t h r o u g h w = w 0 and l e t us c h o o s e that b r a n c h of the r a d i c a l ~ which a s s u m e s a p o s i t i v e v a l u e on the u p p e r e d g e of t h e s l i t in the p l a n e C a~, ~ = w/w0, to t h e r i g h t of 1. T h e n the function

379

W (w) -- ~ 3

I/

u'...~0 W

t --

d_.~.w W

Wo

maps Cw\t on the domain {W:U < O} U {W: 0 < V < 2~} ~

= ~ +iV).

L e t ~2(~) = l o g ~, w h e r e we t a k e t h a t b r a n c h of the l o g a r i t h m which i s r e g u l a r in K 0 = K \ [ 1 , ~] with l o g (--1) = ~ri. T h i s function m a p s K 0 on the h a l f - s t r i p {W:O
o), (z) = VTe-'r

S (z + 2e21r

- e 'r (~ -}- - , ) ) , / 2 d_! '

in which we t a k e t h a t b r a n c h of the r a d i c a l w h i c h i s e q u a l to i e i r (y + T -1 - - 2 COS 2~b)I/2 f o r z = e - i ~ , 0 --< r < 27r. In the c a s e T < 1 the function W = wp(z) m a p s E]~ on the d o m a i n {W:U<0}0iw:u
0
with E~:4~E~(k)/k, E----4~E(k)/k; il~

1

E,(k)= ( t /

/-~2_

E(k)= l "

i

k = 2V;/(' +

i25)

0

a n d in t h e c a s e "r = 1 it m a p s E# on the l e f t h a l f of the p l a n e C W s u c h t h a t the p o i n t s of the unit c i r c l e a r e m a p p e d on the p o i n t of the s e g m e n t of the i m a g i n a r y a x i s f r o m 0 to 8i4T. T h e s e m a p p i n g s a r e c o n f o r m a l a n d u n i v a l e n t in t h e i r r e s p e c t i v e d o m a i n s . 3.

F o r E = 7r t h e r e l a t i o n s

W(w) --ai---- r (z) --iE,

W(w) - - ~ i = ~ i m p l i c i t l y d e t e r m i n e a b o u n d a r y p a i r [fp(z), F p ( { ) ] .

(26)

(~) -- ~i-~-~1

On w r i t i n g the c o n d i t i o n 1~ = +r in the f o r m

a2k~/l 6E2(k) = t,

(27 )

we s e e t h a t t h i s e q u a t i o n h a s a unique r o o t k in (0, 1] f o r 0 < t < ~z/16. F o r t _> 7r2/16 the r e l a t i o n s

W (w) "axi-----(o~(z) - - 4 i ~ W (w) --ai=~(~) --hi

(28)

d e t e r m i n e a b o u n d a r y p a i r [ft~(z), Fp(~)] c o r r e s p o n d i n g to the c a s e T = 1. F r o m (26), w r i t t e n in the f o r m

[

log ~

, -}-

--2VI--W~

on t a k i n g l i m i t a s ~ ~ ~ we g e t e2 exp (~ ~ - ) , F~ = w0 --U o r , b y v i r t u e of (23), (24), a n d (27), F~ = - - e~(~~ By (23) and (20) we g e t 1 - - p ~ f " / q F ~ = tO" + 7 - 1 ) .

5=~e-~(2

380

exp Since g = -

2 [ P 2 f " / q F ' ] [if~f"],

8E2 t) L.

it f o l l o w s that

(30)

In the c a s e t _> v2/16, on w r i t i n g the s e c o n d r e l a t i o n in (28) in the f o r m log [4wi~wo] --2]/l--wo/w = O, a n d t a k i n g l i m i t a s ~ ~ % we g e t F ; = - - e ir Ie~" f -~- ,~.

(31)

Since now 1 - - P 2 f " / q F ~ = 2t, it f o l l o w s t h a t

L e t us o b s e r v e that f " ~ 0, in the c a s e u n d e r c o n s i d e r a t i o n . 4. If f " = 0, then b y (23) we have t(~-+z -!) = 1 and t o g e t h e r with (27) t h i s g i v e s k = 0 s i n c e the e q u a t i o n 8E2(k) = 7r2(2--k 2) h a s no o t h e r r o o t s . C o n s e q u e n t l y , t = 0 a n d P2 = 0. E q u a t i o n s (2i) and (22) a s s u m e the f o r m 1'2 (z)//2 (z) = l l z 2, F'2 (;) IF2(~) = 11; 2. T h e i r s o l u t i o n s a r e the b o u n d a r y p a i r s (az, aeiSC), w h e r e a is a n o n z e r o c o m p l e x n u m b e r and 0 -< 5 < 27r. Now, l e t Pl = 0.

Then p J " + q F ' = 0 and t(T +T -1) = 2. 9

e2

,

If r = 1, then t = ! and E q s . (31) a n d (32) g i v e

o

9

,

Fit = - - e w -T- f ~t, ] ~t -- 2 e - ' r ]~, T h e c a s e 0 < 7 < i d o e s not h o l d b e c a u s e the e q u a t i o n t6E2(k) = ~z(2--k/) h a s uo r o o t . c o n s e q u e n t l y , P2 = q = 0, which is i m p o s s i b l e f o r b o u n d a r y p a i r s . We g e t an a n a l o g o u s s t a t e m e n t f o r q = 0. F(;)

(33) if 7 = 0, then t = 0 and,

H e r e one of the e q u a t i o n s a s s u m e s the f o r m

(Pl] -' "~P~./ " )-~-2p21~'~ = 0 ,

w h e n c e it f o l l o w s t h a t Pt = Pz = 0. 5. Now, l e t i f ( z ) , F(~)] be a f i x e d b o u n d a r y p a i r , w 0' g, and.u be the q u a n t i t i e s in E q s . (21) and (22) c o r r e s p o n d i n g to it, a n d B/~ =f(Et~ ). L e t us e x t e n d the s l i t in the d o m a i n B# b y adding within d o m a i n D the s l i t a l o n g an a n a l y t i c J o r d a n c u r v e f r o m the p o i n t f ( e i r to do. L e t us d e n o t e the d o m a i n so o b t a i n e d f r o m D b y Dp and i t s p r e i m a g e u n d e r the m a p p i n g w = F ( { ) b y K g . L e t us c h o o s e in the p l a n e C w with the s l i t so o b t a i n e d t h a t s i n g l e - v a l u e d b r a n c h of the r a d i c a l ~ 0 / w which is e q u a l to I f o r w = ~o, and in the c o r r e s p o n d i n g d o m a i n of the p l a n e C u , w h e r e u = (1 + ~ 0 ~ - ] ~ w ) / { i - f - ~ w - ) , we c h o o s e t h e s i n g l e - v a l u e d b r a n c h of the l o g a r i t h m with the n o r m a l i z a t i o n l o g (--1) = hi. L e t us s e t r (w) = log [ (t +~q--zL'd~L,)l(t--}~t--wolw)] --2]/t--u,0pv. I n t e g r a t i n g E q . ( 2 1 ) i n the d o m a i n Ep we g e t the r e l a t i o n 4 , i f ( z ) ] = qp(z), w h e r e c ~ ( z ) =

i' V r ~

(z --

tt) (z --

~ -I)

tL

d z / z with t h a t b r a n c h of the r a d i c a l which is p o s i t i v e on the l e f t e d g e of the s l i t in E~ (as s e e n f r o m z = 0) a f t e r #. It f o l l o w s f r o m (22) t h a t i F ( : ) ] : l o g ~+C; H e r e a s i n g l e - v a l u e d b r a n c h of the l o g a r i t h m in K# is t a k e n .

a~Re[q~(z)l

= Re

z

(34)

F o r z = e i0 we have

t//t~ +---L_ 2

and, c o n s e q u e n t l y , Re{r [!(:)]} =Re

q~,(z)=c

on Izl = 1.

381

F r o m what we h a v e s a i d about the p o s s i b l e t y p e s of d o m a i n s we have Re{@ [F(~)]} = e on t ~l = 1 a l s o , and t h e r e f o r e C = c + iX. R e p r e s e n t i n g (34) in the f o r m Log[F(~) ( t @ l / t - - w o / F ( ~ ) ) 2t~,,o] --21/t-- wolF (~) = c+iX, a n d t a k i n g l i m i t a s ~ ~ ~ we h a v e F ' ( ~ ) = woe2ee+iX/4.

A s a b o v e , we g e t

F' ( oo) =--e'~te~e<+'~]'(O)]4, 7(0)=2e-~*(4/k~--2--t/t)]'(O). F u r t h e r we m a k e the s u b s t i t u t i o n s w = w0x 2 a n d z = e i r domains

(35)

2 in (21) and i n t e g r a t e in the r e s p e e t i v e

(36)

Y7

H e n c e it f o l l o w s t h a t c d e p e n d s only on t and ~ and the f o r m of d e p e n d e n c e is the s a m e f o r e v e r y b o u n d a r y p a i r of any f u n c t i o n a l (B~ Since • w h e r e • (0) = 0, is a h o l o m o r p h i c function, on e q u a t i n g the f r e e t e r m s in the e x p a n s i o n s of the two s i d e s of (36) in the n e i g h b o r h o o d of the p o i n t n = 0 we g e t

t =

r

l,"' l -•

1 -

+~-~)

+

[

"

. j x,

\0

T h i s r e l a t i o n , e q u i v a l e n t to (27), e s t a b l i s h e s a o n e - t o - o n e c o r r e s p o n d e n c e b e t w e e n ~-E (0, 1) and tE (0, 7r2/16). L e t us c o n s i d e r t h e e a s e ~ = 1 c o r r e s p o n d i n g to the c a s e s b) a n d e) of P a r a g r a p h one of the p r e i m a g e s of the p o i n t w = w o. T h e n , a s b e f o r e , we h a v e

t' i

:

jo("r

'-

~"

V~7

d • -=

2r

-

--1

,

1.

L e t z = e io b e

,

and, c o n s e q u e n t l y ,

t=~2/16 cos 2 ~12(e--t~). T h i s m e a n s that t _>_7r2/16 in t h i s e a s e . Thus,

it i s p r o v e d that (35) h o l d s f o r e v e r y b o u n d a r y p a i r of the f u n c t i o n a l 03 ~ with c d e p e n d i n g only

on t. 6. L e t !gl~ d e n o t e the f a m i l y of the p a i r s [af#(z), a F # ( ~ e i 6 ) ] and (az, a e i 6 ~ ) , a~C\{O}, 0 <-6 ~. 27r. We f o r m the f a m i l y ~B f r o m it w i t h the h e l p of (3). L e t the f u n c t i o n a l (]3~ with the f u n c t i o n J and the f u n c t i o n a l B with the function J0 c o r r e s p o n d to e a c h o t h e r u n d e r t h e t r a n s f o r m a t i o n (3). S i n c e I~(0) ,]~=/0(zo)= ( l = - ~ . = ~)

Io = Io (Zo) -

(!" ((,) - - 2:-.i' (,*)) F 0 = F ' . ( t _ i:oi~)~ ,

it f o l l o w s t h a t x 0J~ P~ =

"

-O- J .

p,=xoio

- OJ,-~

p~

oi-~ F x 0/0 --~ -----I=~ 1 -

OJ,

p~

' +x~_(t_~ol=)~ (2" 0

,

2zop~ (1 --I:0lD'-" g OJo.

qo=

oFo

_}_

7 oJ.

=

q,

~

w h e r e Pl, P2, and q a r e g i v e n b y (19). H e n c e it f o l l o w s t h a t the b o u n d a r y p o i n t s of the f i r s t k i n d of the c o r r e s p o n d i n g f u n c t i o n a l s (13~ a n d 03) t r a n s f o r m into e a c h o t h e r u n d e r the t r a n s f o r m a t i o n (3). Thus the following theorem THEOREM

382

2.

The pairs of

is proved. ~B

solve the problem

B on ~. For them

zo) (;~>0), ~'(oo)=a, /"(z0) =a~g~ , o(k, z0), i ' (~o)=a@~(k, " where (])6(k,

go) = ! l~---r 2

i

-~

:i21r

ei~

ra

_

I

-

64

W-'k2e~-,

k > t;

r2

-

'

~k2

r --lz01, 0 ~ 8 < 2 a ,

a ~ C\{0}.

O~qD<2a,

U s i n g the c o n n e c t i o n b e t w e e n N a n d S ' , we e a s i l y e s t a b l i s h t h a t the d o m a i n s of v a l u e s of the f u n e t i o n a l s

s=](~, ~-, ~, ~-),~=f'(~o)/F'(oo), ~----f(zo)/F'(oo), on the class ~'I and of the functional

S=](l"(Zo),/"(zo), l'(Zo), !'(zo)), ZonE,

(B')

on S" (with the s a m e f u n c t i o n J) c o i n c i d e . C O R O L L A R Y . The f a m i l y ~lB i n d u c e s a s u b c l a s s S B of f u n c t i o n s of S" which i n t r o d u c e b o u n d a r y p o i n t s of t h e f i r s t k i n d of t h e f u n c t i o n a l (13") and f o r which

j"(z0)=m0,o(<

zo),

!'(~o)=o,(< zo) (t<~o)

in the n o t a t i o n of T h e o r e m 2. R e m a r k 1.

If the f u n c t i o n a l (B ~ h a s the f o r m S~-](]"(O),

then k = 4 / ~ and, m o r e o v e r ,

(38}

i"(O), F ' ( c o ) , F ' ( c c ) ),

t h e r e e x i s t s a s u b f a m i l y of 9:~ ~' B with I. . .(. 0. ) = e . ,.e ~i e-, ~ F'(oo) = a , a e: c \ ' . ~ } , o.<--~<2~,

s o l v i n g o u r p r o b l e m f o r the f u n c t i o n a l (38). R e m a r k 2. No s u r v e y of the p r o b l e m of f i n d i n g the p a r t L j ' of the b o u n d a r y OOj in the g e n e r a l c a s e h a s b e e n m a d e . . H o w e v e r , in c o n c r e t e e a s e s it is r e d u c e d to the c o n s i d e r e d p r o b l e m s . F o r e x a m p l e , for the funct ional

S= (-l+ if(O) l)l( i + IF'(~) 1) +i( l + t i"(O) I)/( l + i F' (co) I) the p a i r s of ~ ' i n t r o d u c e a l l the p o i n t s of the d o m a i n of v a l u e s of the f u n c t i o n a l I = i f ' ( 0 ) / F ' ( ~ o ) [ . + t ] f " ( 0 ) / F ' ( m ) i on ~ . Simllar cases where unexpected situations arise are shown by Example 2, considered below~ w

EXAMPLES

In this section J = u + iv, r : I z 0 i , o =lg0!, z0EE, g0~Ko I , From Theorem I we immediatelyget the following statement: The closed disk IJl <- Rr, p, where Rr, # is given by (16), is the domain of values of the functional J =f'(zo)/F'(go) on the class ~ . Let us obse1~r that Rp = 4p ~ +i) e-rr/(p--1) for i < p _ P0 and Rp - - i as p ~oo For f'(0) = i we have the ring

i/Rp _< Ill < ~

as the domain of values of the functional I = F~(g) on Z ' .

2. Let J =arT+by , ~ =f'(0)/F'(p), a n d v = (1+if'(0) [)/(1+]F'(p)[). For b the closed set

>aRp >0

the d o m a i n G j [s

{ : = t~1 <~ ~R,o, b - - l,/ ~'-Rff --~"-<~ ,, < bRo ~ l i ~-R<,'- - - ~ -2,, CO alg

U J:O~b,l~:i<

-

)

-

383

I)!

:

1

/

" 1

/

u~ c o

u~-

Fig. I (Fig. 1).

I

u~ V~

Fig. 2

M o r e o v e r , L j c o n s i s t s of only two p a r a l l e l s e g m e n t s on 3Gj.

3. Starting f r o m the inequality l f ' ( 0 ) / F ' ( p ) l ~ Rp, with the help of the t r a n s f o r m a t i o n s [ F ( z - 1 ) - t , f ( ~ - l ) -l] and (3) we get s u c c e s s i v e l y the inequalities I f2(z0)/f'(0)F' (oo) I ~ r 2 R l t ~ [ ( i - r 2 ) , d e s c r i b i n g the d o m a i n s of v a l u e s of the c o r r e s p o n d i n g f u n c t i o n a l s on

9)1. In the following e x a m p l e s

~ = I]'(zo)lF'(~) I, 7 = R e ( i §

~= If'(zo)te'(~)I,

"

.

2;

0.f (z0)l/'(z0)), ff

and in the notation of T h e o r e m 2 we will have @0 = @0(k, r), ~0, ~o = ~0, ~o(k, r ) . 4.

L e t J = a + i~.

T h e n the d o m a i n of v a l u e s of this functional (Fig. 2) is b o u n d e d b y the c a r v e s F+:/~--To. 0+ir

0~k~,

F: : I - ~ T0. ,~+iO0, 0 ~ k ~ ko, and the i m a g i n a r y a x i s . In the e a s e w h e r e 0 < r <_ 2 -- 16/~r 2 (the d o m a i n G~) k 0 is the only r o o t of the e q u a tion 8E2(k) = 2~r2--(1 +r}'2)v2/k 2 on (0, 1]; m o r e o v e r k 0 --" 0 a s r ~ oo But if 2 - - 1 6 / z r 2 _< r < 1 (the domain G~)j then k 0 = 4/7r 24~-~--r. As a c o n s e q u e n c e we obtain the domain of v a l u e s of the functional J =f"(z0)/F'(oo) on !~: I J I - < 21e2[(2~+r)21(l--r2)2]. r :~ is an a r c of the p a r a b o l a 7:~: u = v [(2r + 4 ) / ( 1 - - r 2) T e2v/2]I f o r k ->I. U,

-=

8 u,, UF -- (1--r2y-':ff-e "-' r ~ (2-72 \( ' ~2 )~ r ~ '~' u, -- ( 1 -2rr'-')~' u~ = --~ . U,

-~

"2

2 ~=r

"-zY e " I --r~

~

V,

--

1

I --r

21 DO ~

8 V,,v 1

- -e~ '

6~ V,. -~

&i"

I G~. -j~

vo_p

~,vouo Fig. 3

384

In F i g . 2 we have

9

----

/

~#

u~

Fig. 4

V0

uo

;

5. L e t us c o n s i d e r t h e f u n c t i o n a l J = fi + iT. I t s d o m a i n of v a l u e s i s s y m m e t r i c a l with r e s p e c t to the s t r a i g h t l i n e v = ( l + r Z ) / ( 1 - - r 2) ( F i g . 3 ) a n d is b o u n d e d b y t h e c u r v e s

A+:J:Oo+i(i+r~P'o.o/Oo), A-:J=q)o+i( l+r'q% .~/@o), O~h'
A • a r e s e g m e n t s of t h e s t r a i g h t l i n e s 5 ~. v = (1 + r 2 4- 4 r ) / ( i -- r ~-) u r e 2 u / 2 f o r k >_ 1.

U~ - -

I

i

8

e -- r2~ /t~ = "--7"

Uo ' ul

~

1"- r ~ u f : = v o-I- 4r z~~ t r 2' _ ~

~

8

bf'~ t ~ , ~

t--

,

o --2--- -

V:

4r--

e~

I - - , "2

r(l--rD

"

'

l ~ r ~ +4r --

[ --

rz

On t h e b a s i s of t h i s r e s u l t we can s o l v e on ~ the p r o b l e m on the c o n v e x i t y of the l e v e l line L ( r ) of t h e f u n c t i o n f ( z ) in t e r m s of I f ' ( z ) / F ' ( ~ ) l . If t h e i n e q u a l i t y a ( r ) / ( 1 - - r 2) <_ I f ' ( z ) / F ' p o ) l _< 1 / ( 1 - r 2 ) , w h e r e c~(r)-- 2 [ 4 - - , r @ r - - l ) ] ~ l i m : ~ ( r ) = ~ / e - , e2

'

r~I

is s a t i s f i e d on Izl : r f o r i f ( z ) , F ( [ ) ] 6 !I~, then t h e l e v e l l i n e L ( r ) i s c o n v e x . 6.

L e t J =/3 + i y / r / 3 .

If 0 -< r < 2 -- 4"if, then we o b t a i n an u n b o u n d e d d o m a i n of v a h e s

G~T b e t w e e n the

c u r v e s

A t - : J = O o - T i ~ ,O, o + ~ % . , rO~

=

A-:J

9 o+

i|

r~0,.~

r|

k ~[O,

co).

(39)

In t h i s case F' (oo) ~ [

:.i" (z,)~

~,~

[ E, (k)~ k2

8

F o r 2 - 4 3 "<- r < 1 the d o m a i n G j is b o u n d e d b y the c u r v e s (39) and t h e i m a g i n a r y a x i s . The c o r r e s p o n d i n g c a s e s a r e r e p r e s e n t e d in F i g . 4, w h e r e the n o t a t i o n of E x a m p l e 5 i s u s e d f o r the p o i n t s on the r e a l axis, and moreover

I,o = r + r-'~ v o - - TeV~-o , vi = = 2e2- [ T2 :~"- (r + r-~--: 4) :? I ] . A • is an a r c o f t h e h y p e r b o l a X=L:v = ( r + r -1 •

1 / u ; e2/2 f o r k _ 1.

Remarks. 1. It is e a s i l y shown f o r the f u n c t i o n a l s c o n s i d e r e d a b o v e ( e x c e p t the c a s e f ' ( z 0) = 0 of E x a m p l e 4) t h a t a l l t h e i r n o n s i n g u l a r b o u n d a r y p o i n t s a r e of t h e f i r s t k i n d . 2. In E x a m p I e s 4, 5, and 6 we e a s i l y c o n s t r u c t with the h e l p of the c u r v e s 7% 6 +, a n d X• m a j o r i z t n g d o m a i n s w h o s e k n o w l e d g e a l l o w s us to o b t a i n s i m p l e e s t i m a t e s w h i c h a r e e x a c t within d e f i n i t e l i m i t s . A s s u m i n g F ' ( ~ ) = 1 in t h e s e e x a m p l e s , we g e t the s t a t e m e n t s f o r the c o r r e s p o n d i n g f u n e t i o n a l s on the cIass S'. 3. T h e f o r m u l a (d/dk)@0(k, r) = ~0(k, r ) . ( ~ 2 _ 4 E K ) / 2 k E 2, o b t a i n e d with the h e l p of the L e g e n d r e r e l a t i o n f o r c o m p l e t e e l l i p t i c i n t e g r a l s , is u s e d f o r the i n v e s t i g a t i o n of the c o n c r e t e c a s e s of the p r o b l e m B . LITERATURE

Io 2. 3.

4, 5. 6.

CITED

G . M. G o l u z i n , G e o m e t r i c T h e o r y of F u n c t i o n s of a C o m p l e x V a r i a b l e [in R u s s i a n ] , Nauka, M o s c o w (1966). N. A . Lebedev, "On the domain of v a l u e s of a functional in a p r o b l e m on n o n o v e r l a p p i n g domains," D o k l . A k a d . Nauk SSSR, 115, N o . 6 , 1070-1073 (1957). G . V. U l i n a , "On d o m a i n s of v a l u e s of c e r t a i n s y s t e m s of f u n c t i o n a l s in c l a s s e s of u n i v a ! e n t f u n c t i o n s , " V e s t n i k L e n i n g r a d s k o g o U n i v e r s i t e t a , S e r . M a t e m a t i k a , M e k h a n i k a i A s t r o n o m i y a , 7, N o . l , 3 5 - 5 4 (1960). J . A . J e n k i n s , "A r e m a r k on ' p a i r s ' of r e g u l a r f u n c t i o n s , " P r o c . A m e r . M a t h . S o c . , 31, 1 1 9 - t 2 1 (1972). J . A . J e n k i n s , U n i v a l e n t F u n c t i o n s a n d C o n f o r m a l M a p p i n g , S p r i n g e r - V e r l a g , B e r l i n (1958). R. KfJhnau, " U b e r zwei K l a s s e n s e h I i c h t e r k o n f o r m e r A b b i l d u n g e n , " M a t h . N a c h r . , 499, N o s . 1-6, 173185 (1971). 385

G. D. Suvorov, F a m i l i e s of P l a n a r Topological Mappings [in Russian], !zd. Sib. Otd-niya, Akad. Nauk SSSR, N o v o s i b i r s k (1965). N. A. Lebedev, "A m a j o r i z i n g domain for the e x p r e s s i o n I = In (zXf'(z)~-k)/f(z) k in the c l a s s S," Vestnik Leningradskogo Universiteta, Set. Matematika, Fizika, Khimiya, 8, No.3, 29-41 (1955). I. A. Aleksandrov, HExtremal p r o p e r t i e s of the c l a s s S(w0),' Trudy Tomskogo Universiteta, Q u e s tions of the G e o m e t r i c T h e o r y of Functions, 169, No. 1, 24-58 {1963).

7.

8. 9.

INVERSE OF

SEMIGROUPS

UNIVERSAL D.

A.

OF

LOCAL

AUTOMORPHISMS

ALGEBRAS UDC 519.4

Bredikhin

Throughout this a r t i c l e , a u n i v e r s a l algebra has the meaning of a set with a collection of finitary operations. A local a u t o m o r p h i s m of a universal albegra ~ is defined by an i s o m o r p h i s m between its subalgebras (including the empty subalgebra if ~ does not contain nullary operations). The set of all local a u t o m o r p h i s m s of the albegra ~ f o r m s an inverse semigroup denoted by La',~). The semigroup L a ( ~ ) c o m p r i s e s many of the f e a t u r e s to be used in the study of the algebra ~ . Thus the s t r u c t u r e of the subalg e b r a s S u ( ~ ) of the algebra ~/~ is i s o m o r p h i c to the s t r u c t u r e of the idempotents of L a ( ~ ) , and the s e m i group of m o n o m o r p h i s m s Mon(~) and the group of a u t o m o r p h i s m s A u t ( ~ ) are subsemigroups of La(~//). The principal aim of this paper is to find a concrete c h a r a c t e r i z a t i o n of the s e m i g r o u p s of all local a u t o m o r p h i s m s of a l g e b r a s (unary algebras).* The obtained r e s u l t s a r e used for seeking: a) An a b s t r a c t c h a r a c t e r i z a t i o n of the s e m i g r o u p s of all local a u t o m o r p h i s m s of algebras {unary algebras). b) A joint concrete c h a r a c t e r i z a t i o n of Sn(~?1) and .Mon(~//). c) A c o n c r e t e c h a r a c t e r i z a t i o n of Mon(~) and a number of r e s u l t s that strengthen the r e s u l t s obtained in [1-4]. An a b s t r a c t c h a r a c t e r i z a t i o n La(~/) equivalent to the one obtained here has been obtained for the f i r s t time by Domanov [5}. w A partial t r a n s f o r m a t i o n of the set X is specified by a single-valued binary relation defined on the set X. We shall use the following notation: ~(X) is the set of all subsets of X, JX the set of all partial o n e - t o one t r a n s f o r m a t i o n s of the set X, Ax the set of all o n e - t o - o n e t r a n s f o r m a t i o n s of the set X, and ~ , the set of all substitution of the set X ( i . e . , of bijections of X into itself). --1

--!

Let us write hx~--{(x, x) tx~X}, q~={(z, g) l(g, x)~qD}, {(x, g)](:4z) (x, z)~r and (z, g)~ll~} , where % $~J~. Definition 1.1.

A subset

pr~q~{x](~y)(x, g)~q~}, pr2,~--t-'r

W e J, is called a chain if for any q~, r

Definition 1.2. A subset W~Jx is compatible if U W ~ J x . patible on a s e t Y c X if Y c p r ~ ( ~ l - I ~ ) .

we have ( ~

~~

or ~1~r

The t r a n s f o r m a t i o n s q~, ~ J x

are c o m -

Let us recall that a s y s t e m of c l o s u r e s is algebraic if the closure of any set is equal to the union of c l o s u r e s of all its finite s u b s e t s . A s y s t e m of c l o s u r e s is inductive (topological) if it is closed with respect to a union of nonempty chains {arbitrary families) of closed subsets. * The c o r r e s p o n d i n g p r o b l e m for a u t o m o r p h i s m groups is considered by Jonsson [1]. T r a n s l a t e d f r o m Sibirskii Matematicheskii Zhurnal, Vol. 17, No.3, pp.499-507, May-June, 1976. Original article submitted October 24, 1974. I

l 386

This material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 17th Street, N e w York, N. Y. 10011. N o part' I of' this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, ptTotocopying, I microfilming, recording or otherwise, w i t h o u t written permisMon o f the publisher. A copy o f this article is available from the publisher for $ 7.50. [