(A1) Some problems related to iterative methods in eonformal mapping ARKIV FOR MATEMATIK Band 7 nr 8

ARKIV FOR MATEMATIK Band 7 nr 8

1.67 04 2 Cortmlunicated 22 February 1967 by T. NAOET.L and L. CARLESO~r

S o m e p r o b l e m s r e l a t e d to iterative m e t h o d s in e o n f o r m a l m a p p i n g

By II~GEMAR LIND

Introduction

A. The conformal m a p p i n g problem for domains of c o n n e c t i v i t y greater t h a n one has been a t t a c k e d in several ways. The desired m a p p i n g function can sometimes be f o u n d as the solution of a n e x t r e m a l problem or of an integral e q u a t i o n or its existence m a y in certain cases be p r o v e d b y means of the m e t h o d of continuity. A n o t h e r method, sometimes called the /unction-theoretical iteration process, is to express t h e m a p p i n g function F(z) as a composition of functions {/n}7:

~ ( Z ) = /n(/n-l(... (/I (Z)) .-.));

(A1)

F(z) = lim F~ (z),

n - - ~ o o

where t h e / n (determined in some way, e.g. see Hiibner below) are m e r o m o r p h i c a n d univalent in certain simply connected domains. A n a d v a n t a g e of this m e t h o d is t h a t it connects t h e theoretical a n d the constructive questions a b o u t the mapping.

Hiibner ([10] pp. 43-55) has c o n s t r u c t e d a process--the general iteration process-- b y means of which e v e r y function F(z) con_formally m a p p i n g one d o m a i n onto a d o m a i n with analytic b o u n d a r y can be expressed according to (A1). Thus, theore- tically several of the well-known canonical mappings can be expressed in this way.

However, t h e d e t e r m i n a t i o n o f / n , n = 1, 2 ... as a rule requires knowledge of F(z) itself a n d t h u s the process from a constructive point of view has little interest.

B u t there exist exceptions, n a m e l y the circular ring m a p p i n g a n d the m a p p i n g o n t o the lemniscate domain, studied earlier b y Walsh, G r u n s k y a n d L a n d a u .

A detailed a c c o u n t of the problems referred to in this section a n d the following is f o u n d in [3] pp. 208-240.

B. A s t r a i g h t f o r w a r d a t t e m p t to use the function-theoretical iteration process is sketched below. F o r b r e v i t y we call it the iterative process. H e r e the d e t e r m i n a t i o n of the f u n c t i o n s / ~ causes no trouble b u t on the other h a n d the convergence question is more intricate.

L e t D be a d o m a i n of c o n n e c t i v i t y k >~2 on the z-sphere with the c o n t i n u a C~ (~

v = 1, 2 ... k, as b o u n d a r y components. L e t D (n) = Fn(D) (F,~ to be d e t e r m i n e d later) h a v e b o u n d a r y c o m p o n e n t s C~ (n) corresponding to C~ (~ v = 1, 2 ... k. I t is required to find F(z), e v e n t u a l l y restricted b y some normalization conditions, conformally m a p p i n g D onto a d o m a i n ~ so t h a t C~ (~ corresponds to L~, v = 1, 2 ... k. Here L~

I. LIND, Iterative methods in conformal mapping

is some definite continuum (e.g. the unit circle) or some t y p e of a continuum (e.g. a circle or a slit making a prescribed angle with the positive real axis), v = 1, 2 ... k.

I t is of course to be understood t h a t the mapping conditions are appropriately for- m u l a t e d so t h a t the mapping is not "overdetermined". Then/n+l, n=O, 1, 2, ..., is determined in the following way. According to some rule we choose a natural num- ber r~, 1 <vn < k , and look at the simply connected domain D (n) bounded b y C (n) _{v~ v n} and containing the other components. This domain is m a p p e d conformally onto a domain bounded b y the continuum or the t y p e of continuum L , , associated with C~ (~ Together with suitable normalization conditions this determines /~+i a n d C~ (~+i), v - 1, 2 ... k, as well.

The iterative process has proved successful mainly in two cases, namely (a) the circle domain m a p p i n g and (b) certain simple slit mappings.

Case (a) is due to Koebe [12]. Suppose t h a t the point at infinity belongs to D.

We require t h a t

F ( z ) = z + a i + ... + a , + ... (B2)

Z Z n

be the Laurent expansion in the neighbourhood of infinity. We n o r m a l i z e / n in t h e same manner and for example choose Vn SO t h a t v n - - l - - n (mod ]c), n = 0 , 1, 2 . . . The proof of convergence is based on the following. I t can be shown t h a t every kernel of {D(n)}~ has a specific reflection p r o p e r t y (compare [3] pp. 212-214) and t h a t every domain having this p r o p e r t y is necessarily a circle domain.

Koebe ([13] especially pp. 288-296) also gave another proof for the same theorem using a somewhat different iterative technique ("das Iterationsverfahren" as distinct from the former, called "das iterierende Verfahren") in which in order to obtain quicker convergence the reflection properties are exploited more.

K o m a t u [14] has used the iterative process for the circular ring mapping. Cer- tainly this is a special case of K o e b e ' s general theorem b u t K o m a t u uses a measure of convergence which does not involve reflection.

Case (b) is due to Gr6tzsch ([6], [7], [9]) and p a r t l y to Golusin [4]. The proofs are carried out for the parallel, circular and radial slit mappings. I n the case of a parallel slit domain, where the slits are parallel to the real axis, the process can be outlined as follows: Assuming as before t h a t D contains the point at infinity, we require t h a t F(z) a n d / ~ are normalized according to (B2). Let a(1 n) be the first coefficient of the L a u r e n t expansion of/~ in the neighbourhood of infinity and A(1 ~) the correspond- ing coefficient of F~. Then A(1 n) = ~ / ~ 1 a (') Also v~ m a y be chosen as above (other 1 *

rules of choice m a y however lead to quicker convergence). The proof of convergence is based on the following. If {D (n) }F is supposed to have a kernel which is not a parallel slit domain, then necessarily ~ , = l R e a(i ")-+ ~ . On the other hand, it is clear t h a t I A(I")I is uniformly bounded and this gives a contradiction. The well-known extre- mal p r o p e r t y of the m a p p i n g in question is an immediate by-product of this. The circular a n d radial slit mappings arc treated in a similar way. Gr6tzsch also gives estimates concerning the rate of convergence especially in [9].

C. The vital point of the proofs of Gr6tzsch and Golusin above is the v e r y simple behaviour of the functionals a(i n). I t seems reasonable to suppose t h a t proofs based on less special properties m a y have wider scope. A similar r e m a r k can be m a d e a b o u t Koebe's proof.

ARKIV FOR MATEMATIK. Bd 7 nr 8 The following p a p e r mainly deals with constructions of such alternative proofs, discusses some possible extensions a n d in certain cases carries t h e m through.

I n

Chapter I

is introduced a modified iterative process in which every step in- volves a conformal m a p p i n g of a ( k - 1 ) - c o n n e c t e d d o m a i n - - t h e original domain being k-connected. I t is proved t h a t this process can be successfully applied to a n u m b e r of slit mappings. The proofs of convergence are essentially dependent on the m a x i m u m principle for harmonic functions. Besides two examples of extremal properties cormected to the mappings referred to in this p a r a g r a p h are proved with the aid of the iterative process. We conclude the chapter with an estimate of the rate of convergence of Gr6tzsch's process.

I n

Chapter I I

the iterative process is applied to the case where the b o u n d a r y components of s (see B) are required to be simple, closed and analytic, for the rest t h e y are of arbitrarily prescribed types. I t is proved t h a t the process converges if D is subjected to certain geometrical conditions, depending on ~D and ~s a n d essen- tially meaning t h a t the b o u n d a r y components are sufficiently separated from each other.

Chapter I I I

deals with two eigenvalue problems, which m a y h a v e a certain inte- rest in themselves. Some partial results are deduced and the connection between these problems and the classification problem of certain R i e m a n n surfaces is dis- cussed. Especially this problem is studied for a surface, which can be associated with a circle domain and which also plays an i m p o r t a n t p a r t in K o e b e ' s proofs referred to above.

The aim of

Chapter I V

is to discuss an application of the process of Chapter I in the case when s is a mixed rectilinear slit domain of connectivity 2, t h a t is its b o u n d a r y consists of two rectilinear slits making a non-zero angle with one another.

D is supposed to be of "nearly right" shape. We state a sufficient convergence con- dition which is connected with the eigenvalue problems of Chapter I I I .

Professor L e n n a r t Carleson suggested the subject of this paper. I wish to express m y gratitude for his generously given advice and kind interest in m y work.

I. C o n f o r m a l m a p p i n g s o n t o c e r t a i n d o m a i n s o f slit t y p e

1. Definitions

B y a k-connected domain D we shall in this p a p e r understand a domain bounded b y k disjoint continua C1, C,, .... 6~, 0D ⁼ [.iv=iCy . k

Let D be a domain on the z-sphere. Then we m a k e the following definitions.

De/inition

1.1. Y,(D)

is the class o//unctions/(z) meromorphic and univalent in D which have the Laurent expansion

/(z) =z+a~ "'" + ~ + " "

in the ncighbourhood o/ infinity. It is to be understood that D contains the point at in/inity.

103

I. LIND, Iterative methods in conformal mapping

Definition 1.2. E ' ( D ) is the subclass o / E ( D ) consisting o/those/unctions/or which a o = O.

Definition 1.3. E0(D ) is the subclass o/ E(D) consisting o/those/unctions/or which /(0) = 0 . It is to be understood that D in this case contains the origin.

2. The modified iterative process O u r aim is to express a canonical m a p p i n g F(z) in the f o r m

zn(z) = z n ( z ~ - l ( . . . (zl(z)) ...)); F(z)= lim z~(z) (2.1)

n - - > ~

where zn(z~_i), n = l , 2 ... (z0=z), is a conformal m a p p i n g of a certain ( k - 1 ) - c o n - n e c t e d d o m a i n on the z~_l-sphere--the original d o m a i n being k-connected. W e write z~(zm) =z~(z=_l(...(zm)...)),n>m, a n d d e n o t e the inverse of z~(zm) b y Zm(Z~).

Some formal n o t a t i o n will be used t h r o u g h o u t this chapter. L e t D = D (~ be a k-cormected d o m a i n on the z-sphere a n d let ~D = [3 ~=lCg where Cg = C~ ~ = 1, 2 ...

k, are the b o u n d a r y components. W e write D (~) =zn(D) a n d ~D (~) = U~=lC<ff where C(~) corresponds to C (~ u n d e r t h e m a p p i n g z~(z), # = 1, 2, k. B y D~ ~) is m e a n t the ( k - 1 ) - c o n n e c t e d d o m a i n such t h a t ~D~<~)= [J,.~C~ ~) a n d C~)~D~ ~). Unless other- wise m e n t i o n e d t h e letter k stands for c o n n e c t i v i t y a n d D for domain.

T h e precise description of t h e modified iterative process differs slightly f r o m one case to another. T h e differences h o w e v e r are p u r e l y formal a n d therefore it is suffi- cient to describe a t y p i c a l situation.

Our result is t h a t for a n y finitely-connected d o m a i n D on t h e z-sphere ( ~ E D) there exists F(z)EE'(D) m a p p i n g D o n t o a canonical d o m a i n of some t y p e A (e.g. a parallel slit domain). Our proof runs in three steps. (a) T h e s t a t e m e n t is p r o v e d t r u e for k = 1. (b) Assuming it true for k - 1 we determine z~+i(z~) as follows: L e t z~+l(z~) EE'(D~: )) a n d let it m a p D~: ) onto a d o m a i n of t y p e A, n - 0 , 1, 2 ... Here {v=}~ is a n y sequence of the n u m b e r s 1, 2 ... k such t h a t Vn+l~=v~, n = 0 , 1, 2 ...

T h e n it is p r o v e d t h a t F(z)=lim=_.~ z~(z)EE'(D) exists, a n d hence necessarily m a p s D onto a d o m a i n of t y p e A. (c) B y i n d u c t i o n the s t a t e m e n t is true.

Fig. 1 shows t h e first steps in a modified iterative process in the case k = 3 . T h e canonical d o m a i n in question is a parallel slit domain.

3. Slit mapping theorems

T h e proof of T h e o r e m 3 . 1 - - w i t h some formal differences--is f o u n d in [15]. F o r the convenience of t h e reader it is r e p e a t e d here. Theorems 3.1-3.10 are classical a n d of course there exist several proofs of t h e m , (see e.g. [11]). T h r o u g h o u t this paper a slit is to be t h o u g h t of as h a v i n g two different edges.

Theorem 3.1. For each O, 0 ~ < 0 < ~ , there exists a unique /unction ~%(z)EE'(D) mapping D onto a domain bounded by rectilinear slits making the angle 0 with the positive real axis.

Remark. W e call such a d o m a i n a 0-angled parallel slit domain.

Proo/. Suppose t h a t ~ is a 0-angled parallel slit d o m a i n on the w-sphere a n d suppose t h a t / ( w ) E E ' ( ~ ) m a p s ~ o n t o a n o t h e r d o m a i n of t h e same type. T h e n

A R K I V FOR M A T E M A T I K . B d 7 n r 8

Q~ ~-~ . . . r,- 7

D

_{C fa) ~ f:~) 15l}

C2 - z - - c 2 . ~ Cz

C 3 9 J I C_ ) C 3

Vo--~ -vx: 1 v z : 3 v ~ : 2

7--0 7-I ~ ' Z $ Z 3

Fig. 1.

e-~~ ) -~o)

is analytic a n d b o u n d e d in t~, it is zero at infinity a n d its i m a g i n a r y p a r t is constant on each b o u n d a r y c o m p o n e n t . H e n c e [(c0)-o). T h u s

qJo(Z)

is u n i q u e if it exists.

The function

Cfo(Z)

exists if k = 1; this follows f r o m the g i e m a n n m a p p i n g t h e o r e m plus a n e l e m e n t a r y t r a n s f o r m a t i o n .

A p p l y i n g the process described in t h e preceding section with Z.+l(Zn)E Y/(D~(~ )) we observe t h a t

un(z.)

= I m

e-~~ -z.), n

= 2 , 3 ... (3.1) is h a r m o n i c a n d b o u n d e d in D~(: ), t h a t u.(oo) = 0 a n d t h a t it is c o n s t a n t on each boun- d a r y c o m p o n e n t (not necessarily the same c o n s t a n t on each) b u t one, C~(~_) 1. (It m a y h a p p e n t h a t it takes a c o n s t a n t value on this c o m p o n e n t too b u t in this case t h e p r o b l e m is solved.) Since

un(Zn)

is the i m a g i n a r y p a r t of a b o u n d e d a n d a n a l y t i c function in D~ (n) this means t h a t

un(z.)

a t t a i n s its m a x i m u m a n d m i n i m u m values on C~:) r W r i t e 7 ' . - C~(:_)~, F ~ - C~(: ) a n d

A n = m a x ] I m e - ' ~ m a x l u . ( z ; ) - u . ( z ; ) l .

z;,, z;~ ~ y . z~,, z;i ~ rn

I n particular it follows t h a t

[u.(z.)[ <An, z.eT'.,

F r o m the m a x i m u m principle we deduce t h a t there exists a n u m b e r q., O < q. < 1, such t h a t

m a x

lUn(Zn)-Un(Zn)l~.Anqn.

z;e z;i ~ P.

F r o m [1], Ch. IV, 26E, p. 263, c o m b i n e d with t h e c o m p a c t n e s s of

E'(D),

it follows t h a t it is possible to find a n u m b e r q, 0 < q < 1, such t h a t

qn

can be chosen ~ q for all n~>2.

T h u s m a x

lu,(z~)-u,(zn)l <~A,q.

z;~. z;; ~ P,

I. LIND, Iterative methods in conformal mapping

B u t since I m e-~az,~ is constant on F~ this means t h a t Thus when n-~ c~ we have

A.+ x ~< A.q.

A. =O(q")~O. (3.2)

This implies t h a t limn_~r zn(z) exists. F o r suppose the contrary. Since E'(D) is a compact family it is then possible to select two subsequences converging to different functions ~ol(z), ~o2(z ) EE'(D). F r o m (3.2) it follows t h a t (Ol(D) and ~o2(D ) are both 0-angled parallel slit domains. T h e y are conformally equivalent under a m a p p i n g belonging to E ' an~l thus according to the uniqueness eol(z ) -~%(z) which gives a contradiction. Theorem 3.1 is now proved.

Remark 1. The proof indicates t h a t one can estimate the rate of convergence of the process in terms of quantities which depend only on D. L e t ~ =~o(D) and let

~)n(W)=Zn(~oa(o)))

which thus m a p s ~ onto D ("). F r o m l i m e i~162 ~<A~, t o E , S , it follows t h a t in a n y closed subset A of ~ we h a v e

l%(o~)-co I <KAA,, eoeA,

where K A depends on A and ~) only. The only non-rectilinear b o u n d a r y component of ~D (~) is 7~ = zn(F~-l). Let zn-1 be the reflection of z~_ t with respect to lP~_1 (n ~> 3).

Then we define zn =z~(zn-l(z~)) to be the reflection of zn with respect to ~n" I t now follows from the reflection principle t h a t ~v~((o) can be analytically continued over each of the b o u n d a r y components of ~ onto a suitably chosen k-connected domain G on a many-sheeted R i e m a l m surface branched at the endpoints of the slits of ~ . The continuations are given b y ~(oJ*), where the symbols - and * denote the reflec- tion operators with respect to a slit a n d its image respectively. F r o m the compactness of Y/(~) it follows t h a t ~G can be chosen so t h a t its projection onto the to-sphere is a fixed, closed subset A of ~ independent of n and such t h a t I ~ * - w ] ~<KAn_I, r EA, (K independent of n). Observing t h a t A~_ 1 <~K'q n-I we finally deduce from the m a x i m u m principle for analytic functions t h a t

I~.(o)-o~ I ~<~r ~ee~,

where C and q depend only on D (or ~). Similar r e m a r k s can be made in connection with the following theorems of this chapter.

Remark 2. The argument in the proof of Theorem 3.1 built on [1], 26E, p. 263 and the compactness of VJ(D) will be frequently used in various forms in this chapter.

All of these forms are essentially the same as the following. L e t F be a c o m p a c t family of k-connected domains D on the z-sphere such t h a t (1 ~ ~D is contained in ]z I ~ R , (2 ~ c~ ED, (3 ~ ZDED a n d (4 ~ z D is distant at least d > 0 from__ ~D. F u r t h e r let u(z) be a n y harmonic function in D such t h a t u ( o o ) = 0 a n d limzeDIU(Z)[ = 1 . Then ]U(ZD)] ~< q < 1 where q depends on F a n d d only.

Theorem 3.2. There exists a unique /unction O(z)E~J0(D ) mapping D onto a domain bounded by slits on concentric circles with the origin as centre.

Proo/. The proof is analogous to the preceding one. W i t h the same notation we observe t h a t the branch of log ([(eo)/eo) which is zero a t infinity is analytic in ~ a n d

ARKIV FOR MATEMATIK. B d 7 nr 8

t h a t its real p a r t is constant on each b o u n d a r y component. Thus uniqueness follows.

Prescribing Z~+I(Zn)EEo(D~(: )) we choose the branch of log (z~+l(zn)/Zn) which is zero a t infinity. We can now a p p l y the same argul~ent to

un (z~) = l~e log zn +1 (z~), n = 2, 3 . . . . , (3.3)

g n

as we did to the functions (3.1).

Theorem 3.3. There exists a unique /unction tF(z)EEo(D) mapping D onto a domain bounded by slits on half-rays emanating from the origin.

Proof. The proof is analogous to the preceding one. Here the imaginary p a r t of log (f(~o)/~o) is constant on each b o u n d a r y component a n d instead of (3.3) we use

Zn+l(Zn)

u~ (z~) = I m log - - (3.4)

Zn

Theorem 3.4. For each O, 0 <0 <~r, 0 4=:r/2, there exists a unigue /unction /o(Z) EF, o(D) mapping D onto a domain bounded by slits on logarithmic spirals making the angle 0 with half-rays emanating from the origin.

Proof. The proof is analogous to the preceding ones. I n s t e a d of (3.3) we use un (Zn) = I m e -~~ log z~ +1 (Zn). (3.5)

Zn

Theorem 3.5. There exists a unique/unction (o = Ol(z) conformally mapping D onto a domain contained in Ir [< 1, bounded by I w [= 1 corresponding to C~ and slits on vircles centred at the origin and such that

' >

O~(o)=o, o~(o) o, (OeD).

Proof. The proof (see e.g. [11] p. 74) m a y be based on the possibility of m a p p i n g a certain domain of connectivity 2 ( k - l ) conformMly onto a circular slit domain a n d this m a p p i n g can be represented in iterativc terms according to Theorem 3.2.

T h e proof can be carried through in a direct w a y too. I t is then more suitable to change the conditions so t h a t (I)~(0)= 1 a n d to let the radius of the outer circle be unspecified. Then whenever % = 1 , we define Z=+l(Zn)=~(z~)/O'(O) where (I) is the function of Theorem 3.2 with respect to D (n). The proof is analogous to t h a t of T h e o r e m 3.2.

Theorem 3.6. There exists a unigue /unction ~o =tFl(z ) con/ormally mapping D onto a domain contained in ](o I < 1, bounded by I w ]= 1 corresponding to C 1 and slits on half-rays emanating from the origin and such that

~1(0)=0, W~(0)>0, (0eD).

Proof. The proof (see e.g. [11] p. 74) m a y be based on the possibility of m a p p i n g a certain domain of connectivity 2 ( k - 1 ) conformally onto a radial slit domain a n d 107

I. LIND, Iterative methods in conformal mapping

this m a p p i n g can be represented in iterative terms according to T h e o r e m 3.3. On the other hand, the proof can be carried t h r o u g h in a direct way. As in the preceding proof we change the conditions so t h a t ~F~(0)= 1. F o r k = 2 , ~ = 1/z m a p s D onto D'.

T h e n it is possible to find a 0 such t h a t ~0(~) (Theorem 3.1) m a p s D ' o n t o D" b o u n d e d b y two slits on one a n d the same line. A suitable, e l e m e n t a r y root t r a n s f o r m a t i o n t h e n m a p s D" onto a d o m a i n of the desired type. F o r k > 2 we m a y assume t h a t D is c o n t a i n e d in {z { < r a n d t h a t C1 is the circle [z { = r. I n the iterative process it is t h e n always possible to prescribe vn 4=1, n = l , 2, ..., a n d Z~+l(Z,)~Fl(z~) (/F~(0)=1) with respect to D~: ). Observing t h a t the function

/n(Zn)=logZn+l(Zn~);

/n(O)=O,

Zn

is analytic a n d b o u n d e d in D (n) t h a t its real p a r t is c o n s t a n t on C(~ ~) a n d t h a t its i m a g i n a r y p a r t is c o n s t a n t on each other b o u n d a r y c o m p o n e n t b u t one, ?~, we conclude t h a t Im/n(z~) a t t a i n s its m a x i m u m a n d m i n i m u m on ?n" T h e n t h e p r o o f is analogous to t h a t of T h e o r e m 3.3.

Thereom 3.7. Suppose that 1 E C 1 (analytic) and k >~ 2. There exists a u n i q u e / u n c t i o n o~ = (O2(z) mapping D con/ormally onto a domain bounded by [w[ = 1 corresponding to CI, ] co [ = r < 1 (r m a y not be prescribed) corresponding to C2 and slits on circles centred at the origin, and such that (I)~(1) = 1.

Pros/. The proof of uniqueness is analogous to t h a t of T h e o r e m 3.2.

I n t h e iterative process we choose v~ to be alternately 2 a n d 1. W e identify Zn+l(Z,) alternately with (Dl(z~)/q)l(1) a n d r ) =(P*(z~). H e r e (DI(~) is t h e function of T h e o r e m 3.5 with respect to D(~ ~) in the former case a n d to D1 (~) i n v e r t e d in ~- = 1 in the latter case. T h u s (I)~(zn) m a p s Dx (~) onto a domairt contained in [zn+, > r~+l( < 1) a n d such t h a t C(2 ~ +1) is t h e circle [z,+ 1 { = r~+ 1. F u r t h e r (I)~'(c~) = c~

a n d (I)*(1) = 1. T h e functions

u~ (zn) = R e l o g - - , n = 2, 3 . . . (3.6)

2~ n

being zero at z n = 1 a n d being b o u n d e d a n d harmonic in D~(: ) b e h a v e like the func- tions (3.3). The zn(z) belong to a c o m p a c t f a m i l y of univalent functions. I t follows t h a t e v e r y kernel of {D(n)}~ r m u s t be an annulus with circular slits of the t y p e described in the theorem. T o g e t h e r with t h e uniqueness this implies t h a t (b2(z) = lim~_+r162 zn(z). T h e o r e m 3.7 is proved.

Remark. Unlike the preceding proofs the above is n o t inductive. F o r k = 2 it is essentially the same as t h a t of K o m a t u (see the i n t r o d u c t i o n p. 102).

Theorem 3.8. Suppose that 1 E C 1 (analytic) and k >~ 2. There exists a u n i q u e / u n c t i o n r =~F~(z) m a p p i n g D eon/ormally onto a domain bounded by [co[ = 1 corresponding to C1, [~o[ = r < 1 (r m a y not be prescribed) corresponding to C~ and slits on hall-rays e m a n a t i n g / r o m the origin and such that ~F2(1 ) = 1.

ARKIV F6R MATEMATIK. B d 7 n r 8

Proo/. The proof is analogous to the preceding one. The iterative process is based on the function tFl(~) of Theorem 3.6. As above we use the functions

Zn+l(Zn)

U n ( Z , ~ ) = R e l o g - - , n = 2 , 3 . . . (3.7) Zn

to prove the convergence. I t is to be observed t h a t u~(zn) attains its extremal values on C~(~_) ( k > 2 ; for k = 2 Theorems 3.7 and 3.8 are identical) since log (Z~+l(Z,)/z~) is analytic and bounded in D~(: ) and its imaginary p a r t is constant on C, (~), ~ 4= l, 2.

Theorem 3.9. For each 0, 0 < 0 < 2 z , there exists a unique/unctionPo(z)EZo(D) mapping D onto a domain bounded by slits on con/ocal co-axial parabolas with the origin as/ocus and the axis making the angle 0 with the positive real axis.

Proo]. Let f2 be a domain on the ~-sphere, 0, oo E ~ . B y ~ is m e a n t the two-sheeted covering surface of ~ branched at zero and infinity.

Supposing ~ on the w-sphere and f2' being parabolic slit domains of the t y p e in question conformally equivalent under a mapping/(co) E E0(~ ) we can define a bound- ed analytic function in ~ to take the values

V ~ / ( ~ o ) - Ve - ' ~ o~

at points lying over o). I t is zero at the branch points and its imaginary p a r t is con- stant on each b o u n d a r y component. Hence /(o))-o) which proves the uniqueness.

The theorem is true for k = l (see e.g. [11], pp. 78-80).

The iterative process is constructed in the standard w a y (zn+l(zn)EE0(D~(:))). W e observe t h a t the functions un(P~) taking the values

Im{Ve-*~ - ~ } , n = 2 , 3 . . . (3.8) at points P~ E ~ ( : ) lying over z~, are bounded and harmonic, are zero at the branch points and are constant on each b o u n d a r y component (of ~/),(:)) except two which lie one over the other, with the b o u n d a r y values differing only in sign. Thus the functions u~(Pn) , n = 2 , 3 ... behave like the functions (3.1) a n d it follows as in Theorem 3.1 t h a t every kernel of {D(n)}~ must be a parabolic slit domain of the type in question. F r o m the uniqueness it then follows t h a t Po(z)=lim~_~ z~(z). Theorem 3.9 is proved.

Theorem 3.10. For each a E D, a 4=0, and o~, 0 <o~<7~, there exists a unique/unction r ) mapping D onto a domain bounded by slits on curves belonging to the/amily

G(a) (e(C+u) e-t~ "~- e_(C+tt)e-in)2, co= 7

where t is a real parameter and c a real constant.

Remark. The family of curves of Theorem 3.10 are trajectories of the family of ellipses with loci at 0 and G(a). F o r ~ = 0 we obtain these ellipses and for o~=~/2 hyperbolas with loci at 0 and G(a).

109

~. L~r(n, Iterative methods in conformal mapping

Proo/. L e t ~ be a domain on the ~-sphere, 0, a E ~ ( a # 0 ) . B y (](a) is m e a n t the two-sheeted covering surface of ~ branched at 0 and a. F u r t h e r we write formally

g(~,A) = ~

The uniqueness follows mainly as in the preceding proof. With analogous notation it is possible to define a bounded analytic function in ~(a) to take the values

e '~ log g(/(eo),/(a)!

g(~o, a)

a t points over r being zero at points over cr Since its real p a r t is constant on each b o u n d a r y component it follows that/(o~) ~ w .

I n the iterative process we choose zn+:(z~)E Y~o(D(~:) ) and the associated p a r a m e t e r is a n =zn(a ). The functions un(Pn) taking the values

Re e ~ log g(zn+l (Zn), an+l)

g(Zn, an) , n = 2, 3, ..., (3.9) a t points P E/~ (n)+" " n ~, t+n) then have properties similar to the functions (3.8) and the a r g u m e n t runs as in the preceding proof.

As regards the case k = l , see e.g. [5], p. 128. Theorem 3.10 is proved.

Theorem 3.11. For each a E D (a #0) there exists a unique/unction co = CI(Z ) E F~0(D ) mapping D onto a domain bounded by slits on circles going through 0 and C:(a),

Proo/. Suppose t h a t ~ on the ~o-sphere is a slit domain of the t y p e described in the theorem (associated p a r a m e t e r a) and suppose t h a t there exists /((o)EE0(~ ) m a p p i n g ~ onto another domain of the same t y p e (associated p a r a m e t e r / ( a ) ) . Then

I

_og

go(/(to) - / ( a ) )

is analytic and bounded in ~ a n d I m g(w) is constant on each b o u n d a r y component a n d this implies t h a t / ( w ) =~o. Thus Cl(Z) is unique if it exists.

I n the case k = l we must prove t h a t there exists a function ~o=/($), univalent a n d meromorphic in E:[ ~[ < 1, having a simple pole with residue 1 at the origin, a n d mapping E onto a domain with b o u n d a r y of the following type: L e t a~EE,

~, # 0 (i = 1, 2), :r #~2, with al, a2 otherwise arbitrary. Then the required b o u n d a r y is to be a slit on a circle through 0 = / ( a l ) and/(as). Suppose first t h a t the straight line through ~1 and as contains the origin a n d makes the angle 0 with the positive real axis (0 ~<0 <~r). Then the desired m a p p i n g is

which m a p s E eonformally onto a domain bounded b y a slit on the straight lino through 0 =1(~1) and/(:r I n another case the system

ARKIV F6R MATEMATIK, B d 7 nr 8

[u--0Q [

i = 1,2, w h i c h is equivalent to

~ [~[

u =i+I F

h a s a unique non-zero solution u, [u[ < 1 a n d

i = 1 , 2 ,

U - - ~ U --061

/(~) -

u ~ ( 1 - ~ ) u a l ( 1 - ~al)

is the desired mapping. I f we write T ( ~ ) = ( u - ~ ) / ( u ~ ( 1 - z ~ ) ) we can readily v e r i f y t h a t / ( 1 ~ 1 = l ) is a slit on t h e circle ~o+T(~l) = u -1. This circle contains 0=/(6r a n d T(o,2)-T(oh)=/(o~2) since u T ( a i ) = 1 , i = 1 , 2, according to t h e choice of u above. T h u s the t h e o r e m is true for k = 1.

As regards the iterative process, we h a v e zn+l(zn) EEo(D(~: )) a n d t h e associated p a r a m e t e r is a~ = zn(a). T h e functions

un (z,) = I m log (zn + 1 (zn) ^-- an + 1) Zn

( z ~ - a n ) z~+l(z,) ' n = 2 , 3 . . . (3.10) b e h a v e like the functions (3.3) a n d t h u s together with t h e uniqueness lead to Cl(z ) = limn_,~o z=(z). T h e o r e m 3.11 is proved 9

Theorem 3.12. F o r each a E D (a # 0 ) there exists a u n i q u e / u n c t i o n eo = C2(z ) E Eo( D ) m a p p i n g D onto a d o m a i n bounded by slits on the circles I (eo-C~(a) )/eo [ = c o n s t .

Proo/. T h e proof is analogous to the preceding one. As regards t h e uniqueness we observe ~that with analogous n o t a t i o n Re g(w) is c o n s t a n t on each b o u n d a r y com- ponent.

F o r the iterative process, instead of the functions (39 we use Un (z.) = Re log (Zn+l (zn) - -

an+l)

^{z n}

( z n - a n ) Zn+l(z,J ' n = 2 ' 3 ' " ' "

I n t h e case k = 1 the t h e o r e m is p r o v e d in a w a y similar to t h a t of t h e preceding proof. W i t h t h e same n o t a t i o n we first suppose t h a t I zr = ] a~l" T h e n it is possible

9 < < ~(o+t) ~(o-t) he desired

t o fred 0 a n d t, 0-~0 :z, t # 0 , so t h a t ~ l = ~ e , a~=~e . T h e n t m a p p i n g is

/(~) =~--1 __ ~1-1 _~_ e-2io(~ _ 0~1) '

which m a p s E conformally o n t o a d o m a i n b o u n d e d b y a slit on the perpendicular bisector of t h e straight line segment joining 0 = / ( 3 1 ) a n d / ( ~ e ) . I f ]31[ ~=[~2[ t h e n

U-- ~

U - - ~ I

/(~) ^-

u~(1--45)

U g l ( 1 -- ~ 1 ) is the m a p p i n g where u = re t~176 is the (unique) solution of

e-iq~062 -~ eieP ~1

r - (3.12)

1 + ~x ~2

I. LIND,

Iterative methods in conformal mapping

such t h a t 0 < r < 1. To verify t h e existence of the solution we m a y suppose that, I ~ 1 ] < [~zl; the other case is s y m m e t r i c a l to this. Taking i m a g i n a r y p a r t s of (3.12}

we o b t a i n

I m e ' ~ ( ~ 1 ( 1 - [ ~ [ 2 ) - ~ ( 1 - (3.13) I t is easily verified t h a t there exists ~ satisfying (3.13) a n d such t h a t Re e-~*~2 > 0 since otherwise we obtain a contradiction to ] ~ ] < I ~ l " Choosing ~his value we observe t h a t

+(~)

1 + z ~ e - ~

m a p s I zl < I ~1 o n t o a disc intersecting the real axis along t h e segment 0 < R e o) <

(2 R e e-~r + la212)<1. F u r t h e r r=w(et~51) is real. Hence 0 < r < l . I f T ( ~ ) =

(u-$)/(u~(1-(t~))

t h e n (3.12) is equivalent to [ u ] 2 T ( ~ l ) T ( ~ 2 ) = 1 . I t is readily verified t h a t ]([ ~] = 1) is a slit on t h e circle

~o(t) = - T(zq) + [u [ -le ~t,

0 ~< t < 2z. T h e choice of u implies t h a t

w(t)-/(~)[=[ul[T(g2)[ I ea-[ul_T_(~2) =

~--1(~:1) 1--r lullT( )l.

T h e o r e m 3.12 is proved.

Remark.

L e t t i n g a-~c~ we see t h a t e v e r y kernel of

{CI(D,

a)} is a radial slit d o m a i n a n d t h a t e v e r y kernel of

{C2(D,

a)} is a circular slit domain,

(C,(z, a)=-C,(z),

i = 1, 2). F r o m the uniqueness of the m a p p i n g s (])(z) a n d W(z) of Theorems 3.2 a n d 3.3 it follows t h a t (I)(z)=lima_~r

Cs(z, a)

a n d ~ F ( z ) = l i m a _ ~

Cl(z, a).

L e t t i n g a->0 we see t h a t every kernel of {el(D, a)} a n d

{c2(n,

a)} is a d o m a i n b o u n d e d b y slits on circles R e ei%o -1 = c o n s t . (0~<~<~). I t is readily verified t h a t there exist kernels corresponding to a n y % 0 ~<~ < z .

We conclude this section with a brief discussion on the application of t h e modified itcrative process to the following problem: Does there exist w = / ( z ) E Z ' ( D ) m a p p i n g D o n t o a d o m a i n b o u n d e d b y slits on lemniscates

Iw-/(al)[ ]w-/(au) [

= c o n s t . , where a 1 a n d a 2 are given points in D? If the choices of a 1 a n d a S are restricted in a certain w a y depending only on D t h e n such a m a p p i n g exists.

Suppose first t h a t k = 1. T h e sets E~ = {/(a~)[/EZ'(D)}, i = 1, 2, are certain closed discs ([5] p. 129) a n d let

E={89

i = 1 , 2}. F u r t h e r there exists a closed disc K such t h a t the b o u n d a r y of

/(D)

is contained in K for all

/EZ'(D)

([5] p. 178). W e n o w m a k e the a s s u m p t i o n t h a t K N E = r This condition is for e x a m p l e satisfied if t h e points a~, i = 1, 2, are at a sufficiently great distance f r o m

OD.

L e t b~ E E~, i = 1, 2. We indicate the construction of a f a m i l y F of simple closed J o r d a n curves in t h e w-plane, as follows (Fig. 2). L e t L be t h e perpendicular bisector of the straight line segment t h r o u g h b 1 a n d b~ a n d let K ' be the reflection of K in t h e segment. N o w L divides the plane in the half-planes H 1 a n d H e. W e d e m a n d t h a t

L c H 1

(so t h a t H 1 is closed a n d H2 open) a n d suppose for example t h a t t h e centre of K is in H I. W e define H i to be H 1 U K U K ' a n d let H i be its complement. A curve belonging to F is to consist of a n arc of a lemniscate I w - b x [[ e o - b2 [ = c o n s t . , w E H i , if this does n o t intersect ~H~ a n d such an arc completed b y d r a w i n g a circular are ]o~- 89 1 + b ~ ) I = c o n s t . , w E H~, if the lemniscate does intersect

OHi.

T h r o u g h each

ARKIV FOR MATEMATIK. B d 7 n r 8

• b z

F i g . 2.

p o i n t in t h e plane there passes precisely one m e m b e r of the family. A slit on a curve in F (which in t h e present case m a y be along the whole closed curve) is uniquely d e t e r m i n e d b y three real parameters. A s t a n d a r d application of the m e t h o d of c o n t i n u i t y proves t h a t there exists a unique f u n c t i o n / ( z ; b 1, b2)EZ'(D) which m a p s D onto a d o m a i n b o u n d e d b y a slit on a curve in F (see [8]).

F r o m the choice of K it follows t h a t this slit m u s t be situated within K. F u r t h e r it is clear t h a t /(z0; b 1, b2), zoED, b~C E , i = 1, 2, is continuous with respect to b 1 a n d b e. R e g a r d i n g b[ =/(a~; b 1, be), i = 1, 2, as a t r a n s f o r m a t i o n of (b 1, b2) the condi- tions for a n application of B r o u w e r ' s fixed point t h e o r e m are fulfilled. H e n c e there exist b~CE~, such t h a t b~=/(a~; bl, b2), i = 1 , 2. T h u s our s t a t e m e n t is true for k = l .

If the choices of a~, i = 1, 2, are a p p r o p r i a t e (depending in particular on k, c o m p a r e [11], p. 96) the iterative m e t h o d can be used. The proof of convergence is based on the functions

Zn+l

(~)

a~ n + l )

u= (z=) = Re log 1~ a(n) , n = 2, 3, ...,

i=l Zn -- i

where Zn+l(Zn) E Z'(D~ (:)) a n d a~ n) =z~(a~), i = 1, 2.

As regards the uniqueness we o b t a i n t h e condition

(/(OJ) - - / ( a 1)) (/(~o) - - / ( a 2 ) ) = (o) - - a l ) (o) -- as),

with obvious notation. Since/(co) E Y / i t follows t h a t a 1 + a 2 = / ( a l ) + / ( a e ) . F u r t h e r 1' (o)) [/(w) ^-- 89 +/(a2))] = eo -- 89 1 + a2).

Since 89 1 +a2) E D it necessarily follows t h a t

/(212(al

+a2) ) = 89 +/(a2)). I t n o w easily follows t h a t a l - - a e = / ( a l ) - - / ( a 2 ) . Thus /(a~)=a, i = 1, 2, a n d finally ](r

4. Extremal properties

Most of t h e m a p p i n g s of the previous section h a v e simple extremal properties, w h i c h as a m a t t e r of fact u n i q u e l y characterize them. The modified iterative process m a y enable t h e actual calculation of the extremal quantities in certain cases. W e give two examples of the extremal properties.

I. LIND, Iterative methods in conformal mapping

Corollary 4.1. Maxr~r..(D , Re {e-2~~ {e-2~~ 0~<0<s, where / ( z ) = z + (al/z) + ... and cfo(z ) = z + (ao/z) + ... are the Laurent expansions near the point at in/inity and q)o(Z) is t h e / u n c t i o n o/ Theorem 3.1. Equality occurs i/ and only i/ D i s a O-angled parallel slit domain.

Pros/. Since Z'(D) is compact the existence of a solution within the class is guaran- teed. Thus it is sufficient to prove that Re {e-2i~ with equality if and only if D is a 0-angled parallel slit domain. Supposing this done we let ~ =/(z) = z + (al/z) + . . . map D onto ~ , which is not a 0-angled parallel slit domain, and let ~o =~o(~)=

~+(bo/~)+... be the function of Theorem 3.1 with respect to ~. Then Cfo(/(z))=

z + (a 1 + bo)/z +... E Z ' ( D ) and

Re-2~~ + bo) > R e e-2~~

Thus/(z) cannot be extremal.

The corollary is true for k = 1 which for example can be proved with the aid of t h e area theorem. Supposing it true for k - 1 we express 0So(Z) iteratively as lim~_~ z~(z)

z~ + a(n)~z

according to Theorem 3.1. Writing z~+l(z~)= ,, / n + ... we deduce t h a t

Hence we have

as= a'o

n = O

R e e-~i~ ao = ~ R e e-2~~ a~ ~) >10. (4.1)

n = 0

Clearly equality occurs if and only if all terms are zero, which means t h a t D is a 0-angled parallel slit domain.

B y induction Corollary 4.1 is true for all k.

Remark. As far as Corollary 4.1 is regarded as a purely qualitative statement t h e proof m a y be simplified. Then the only essential points are the compactness of Z'(D) and the t r u t h of the corollary for k = 1. Considering the remark of Theorem 3.1, (4.1) however enables us to estimate the extremal value. For example given D we can find constants B and q < 1 such that Re e-~i~ ~) <~ Bq ~ and hence l~e e-2~~ =

~ = ~ R e e ~*~ The extremal properties of (D(z), ~F(z), /o(z), (l)l(z), ~IZi(z) (compare [11] pp. 72-77) and G(z) ([5] p. 128) of Theorems 3.1-3.6 and 3.10 can be treated in a similar way.

We return to the extremal property of ~0(z) in Section 5.

As a less obvious example we now prove the extremal property of (D~(z) in Theorem 3.7 which in fact also gives an alternative proof of the existence of the mapping.

Corollary 4.2. Let Flu = F12(D ) be the class o//unctions, co =/(z), regular and univalent in n such that C 1 corresponds to {co { = 1 and C 2 to {co { = r < 1 (k >~ 3). T h e n

Max r =

f E FI~

where {w{ = ~ corresponds to C 2 under (I)~(z) and where (I)2(z) is the/unction o / T h e o r e m 3.7. The m a x i m u m is attained/or the/unctions eir(P2(z) (~ real) only.

ARKIV FOR MATEMATIK. B d 7 nr 8

Proo/. F12

is compact which guarantees the existence of a m a x i m u m . We m a y suppose t h a t D is contained in the a n n u l u s r' < ]z] < 1 (C1: ]z I = 1, C2: ]z I = r ' ) a n d t h a t at least one of the remaining boundar3; components is not a slit on a circle of centre the origin. We now a p p l y an iterative process, which mainly is t h a t of T h e o r e m 3.7 b u t with the difference t h a t Z~§ is identified alternately with @l(Z~)/Ol(0) and O~(O)/q)~(1/z,) (p. 107). Let Q~) be the inner radius (with respect to he origin) of the finite domain bounded b y C~ ~) and let ~(n) be the outer radius of the infinite domain bounded b y C~ n), i = 1, 2; n = 0 , 1, 2 ... Then Q~ ~<~) where equality occurs if and only if C~ n) is a circle of centre the origin. Further, we construct the iterative process so t h a t C(~ 2n+l) and C~ 2~) are circles. We denote their radii R2~+1 and r2~

respectively. According to the extremal p r o p e r t y of O1($ ) we obtain

R~_I =512~) >el2 ~)>~ R2~+~,

r ~(2n-1) < ~(2 n - 1)

2 , ~ - 2 = ~ <~r2~, n = l , 2, ....

Thus r2~-+r > r' a n d

R2n+l--->

R < 1 (since R 1 < R 0 = 1). This proves in particular t h a t {z~(z)}~ r belongs to a compact family of univalent functions and as in Theorem 3.7 it follows t h a t every kernel of {D (~)} m u s t be an annulus r < ] w ] < R with circular slits. F u r t h e r r ' < r / R which proves the extremal property. Corollary 4.2 is proved.

A r e m a r k similar to t h a t of Corollary 4.1 concerning the possibility of estimating can be made here.

5. On rates o f convergence

GrStzsch indicates in [9] (for the case of circular slits) a m e t h o d of obtaining esti- mates of the rate of convergence of the iterative method used b y him. I n the present section we give a similar method concerning mappings onto zero-angled parallel slit domains, a method which gives an explicit estimate of the rate of convergence.

This is not exponential (k>~3), contrary to the rate of convergence of the modified iterative process (Theorem 3.1, see R e m a r k 1, p. 106). The precise rate is not known, (compare also [3], pp. 236-238).

Let the width of a slit F be defined as Maxa. b~r I m ( a - b ) and let A~ be the maxi- mal width of the slits {C~n)}~. Each step in the iterative process is determined in the following way. Consider the simply connected domain which is bounded b y a slit of m a x i m a l width (we denote one such slit b y ?n). This domain is m a p p e d onto a zero-angled parallel slit domain. The mapping functions are normalized in the usual manner:

~(n)

Z~+l(Zn)=Zn+ '~1 § . . . . n = 0 , 1,2 . . . Zn

and we write Re

a(1 n)

=~t n for short. We pose the following problem (k~>3):

Given ~ > O, find a number N~ such that A= < e whenever n >~ N~.

Suppose t h a t ?n = C~ (n) (n > no). L e t n* be the greatest index < n such t h a t ?n*-I = C~ (~*-1). Then C~ ~*) is rectilinear and ?~ is the analytic image of C~ =*) under the m a p p i n g

Z~(Zn_I(...(Zn*)...)).

115

L LIND,

Iterative methods in conformal mapping

W e n e e d some n o t a t i o n . W e s a y t h a t

n*

is

the/irst predecessor

of n a n d t h a t n is

the ]irst successor

of

n*.

G e n e r i c a l l y s p e a k i n g n ( < m) is

a predecessor

of m if t h e r e e x i s t s a

chain

of indices n = n 1 < n 2 < ... < n p = m such t h a t n , is t h e first p r e d e c e s s o r of n~+l, v = l , 2, ..., p - 1 . S u c h a c h a i n is s a i d to h a v e

length p.

T h e r e l a t i o n

"n

is a p r e d e c e s s o r of m or a successor of m or e q u a l s m " is a n e q u i v a l e n c e r e l a t i o n . I t is e a s i l y seen t h a t t h e n u m b e r of e q u i v a l e n c e classes is a t m o s t k - 1 . W e d e n o t e t h e m b y E l , E 2 .. . . , Ek 1. W e s a y t h a t An satisfies a

condition o/quotient

or a

Q-condition

if

An. < QAn,

w h e r e Q is a n a r b i t r a r i l y f i x e d p o s i t i v e n u m b e r . To be c o n c r e t e we shall in t h e fol- lowing let Q e q u a l 2 k-1. F o r c o n v e n i e n c e we s u p p o s e t h a t t h e slits {C(~~ (one of w h i c h is r e c t i l i n e a r ) a r e a n a l y t i c a n d t h a t An, n = 0 , 1, 2, ..., are small, w h i c h is g u a r a n t e e d if A 0 is s u f f i c i e n t l y small. I n t h e following t h e l e t t e r C d e n o t e s a p o s i t i v e c o n s t a n t n o t n e c e s s a r i l y t h e s a m e e a c h t i m e i t occurs b u t in a n y case i n d e p e n d e n t of n.

L e m m a 8.1. 1 ~

I / A n satisfies the Q-condition then

# n ~> C A ~ , n = 0, 1, 2, ....

2 ~ .

I n any case it is true that

o o

ft, ~< C(A n log An) 2, n = 0, 1, 2 . . .

v = n

Proo].

W e o b s e r v e t h a t z~+l(zn) , n > n o can be c o n t i n u e d o v e r ~n ----C~ ~) i n t o a d o u b l e - s h e e t e d R i e m a n n surface b r a n c h e d a t t h e e n d p o i n t s of 7~. T h e c o n i n t u a t i o n is g i v e n b y

z~+l(zn*(zn)) = zn + rn +l(Zn), $

w h e r e

zn*(zn)

is t h e m a p p i n g of D (n) o n t o D (n*) etc., a n d " m e a n s r e f l e c t i o n in C(~ n+l) a n d C~ (n*) r e s p e c t i v e l y . B y a n a r g u m e n t s i m i l a r to t h a t of R e m a r k 1 o n T h e o r e m 3.1 (p. 106) we conclude t h a t

Ir*+l(zn) l

<~c(An, +An+l),

zn EL,

w h e r e L is t h e c u r v e

d(zn,

7 , ) = d > 0 t h o u g h t of as l y i n g in t h e s e c o n d sheet. H e r e

d(zn, 7n)

is t h e d i s t a n c e b e t w e e n zn a n d ~n a n d d can be chosen i n d e p e n d e n t l y of n as a c o n s e q u e n c e of t h e c o m p a c t n e s s of E ' ( D ) . If, besides, An satisfies t h e Q-condi- t i o n t h e n also using An+l < 2An (which is a l w a y s true) we h a v e

]r*+l(zn) I ~<CAn,

znEL.

W e shall s t u d y t h e i n v e r s e of

Zn+l(zn)

a n d for s i m p l i c i t y we w r i t e i t as e ) = / ( z ) =

z + a l / z § ....

w h e r e a i = - a ( 1

n).

Since t h e

zn(z )

b e l o n g to t h e c o m p a c t f a m i l y Z ' ( D ) t h e r e is no loss of g e n e r a l i t y in s u p p o s i n g t h a t t h e r e c t i l i n e a r slit C~ (n+l) e q u a l s

{z,+l I I x 1<~2, y

= 0 } ,

zn+ 1 = x § iy.

A t w o r s t t h i s will o n l y cause s i m p l e m o d i f i c a t i o n s of t h e c o n s t a n t s i n v o l v e d . F u r t h e r we w r i t e A i n s t e a d of A,.

AI{KIV FSl{ MAT]~liIATIK. Bd 7 nr 8 According to t h e a b o v e t h e function

is m e r o m o r p h i c (singular p a r t ~-1) in a fixed d o m a i n containing some closed disc I~l <~<RI" Further(R>l andwe haveiS i n d e p e n d e n t of A, i.e. of n ) a n d it is, m o r e o v e r , u n i v a l e n t i n

} I m h(~)l ~<A, I~[ = 1 , (5.1)

a n d there exists some p o i n t ~0, I~01 = 1, such t h a t

I Ira h(~0) l >~ A/2. (5.2)

If, m o r e o v e r , A n = A satisfies the Q-condition t h e n

Ih(~)l <cA, I$l =R. (5.3)

T h e function g(~) m a p s I ~ 1 < 1 onto a d o m a i n which has a c o m p l e m e n t of zero area. H e n c e we o b t a i n f r o m the area t h e o r e m t h a t

cr

--21:r ~

(5.4)

W e n o w p r o v e 1 ~ I t follows f r o m (5.2) a n d (5.3) t h a t there exists a n arc ~ of length ~ (~ i n d e p e n d e n t of A) on the unit circle such t h a t

[ I m h($)l >~A/3, ~Ea.

(5.5)

W e h a v e

r = l ~ = 1

a n d hence f r o m (5.4) a n d (5.5)

- 2 R e a I ~ - ~ A 2.

9 This p r o v e s 1 ~

W e n o w t u r n to 2 ~ which we p r o v e b y induction on k. I f k = 1 t h e n we use t h e o b s e r v a t i o n s m a d e a b o v e a b o u t the f u n c t i o n / ( ~ ) (here r e g a r d e d as the inverse of

zl(z)).

I t follows f r o m (5.1) t h a t [a~[ ~<2A a n d f u r t h e r we h a v e

[av] <~CR-",

v = l , 2 ... Choosing N = [log 2AC-1/log R -1] we o b t a i n

v = l I N + I N + I

H e n c e f r o m (5.4) we deduce t h a t

- 2 R e a 1 < C ( A log A)L

T h e c o n s t a n t C depends of course on the p a r a m e t e r s d e t e r m i n i n g D b u t C is uni- f o r m l y b o u n d e d as soon as t h e p a r a m e t e r s are s u i t a b l y bounded.

9:2 117

i,

LIND, Iterative methods in conformal mapping

T h u s 2 ~ is t r u e if k = l . U s i n g (4.1), Section 4 (p. 114) we can n o w p r o v e i t t r u e for/C if it is t r u e for/c - 1. I t is e s s e n t i a l h e r e t h a t t h e w i d t h s in t h e m o d i f i e d i t e r a t i v e process decrease e x p o n e n t i a l l y a n d t h a t E ' ( D ) is c o m p a c t . T h e d e t a i l s of t h e p r o o f a r e t h e n e l e m e n t a r y . L e m m a 5.1 is n o w p r o v e d .

L e m m a 5.2. 1 ~ We have SUpm~>n Am ~< CAn ] log

An[, n=0, 1, 2

. . . 2 ~ Let n' be the smallest index > n such that A n, < 1 i n. Then

n ' - n < ~ C (log An) z , n = 0 , 1, 2 . . .

Proo/. F r o m t h e c o n v e r g e n c e of t h e process it follows t h a t SUpm~>n Am is a t t a i n e d for s o m e (smallest) m, a n d we m a y s u p p o s e t h a t m ~> n § since o t h e r w i s e A m < 2kAn.

F u r t h e r A m _ , ~ 2 ~Am, v = 0 , l , 2 . . . I f a n y one of t h e w i d t h s Am_T, V = 0 , 1, ..., k - l , satisfies t h e Q - c o n d i t i o n i t follows f r o m L c m m a 5.1 t h a t

C(21 kAm)2 < ~/z~ < C(A n log An) 2, (5.6)

y = n

a n d 1 ~ follows. S u p p o s e t h a t n o n e of t h e s e w i d t h s satisfies t h e Q-condition. T h e n t h e first p r e d e c e s s o r s ( n > n 0 ) of m, m - I . . . m - k + l a r e all < n since A(~ ~).>~

QA~21 k - - A ~ for v = 0 , 1 . . . / c - - l , ( Q = 2 ~ - 1 ) . T h u s t h e /c c o n s e c u t i v e indices m, m - 1 . . . m - k + l > n all h a v e p r e d e c e s s o r s < n . B u t t h i s is impossible. T h u s (5.6) is t r u e a n d we h a v e p r o v e d 1 ~

To p r o v e 2 ~ we m a k e t w o o b s e r v a t i o n s . F i r s t s u p p o s e t h a t we h a v e a c h a i n n ~< n 1 < n 2 < . . . < nv < n ' such t h a t An, satisfies t h e Q-condition. T h e n we h a v e

W e p r o v e this b y i n d u c t i o n . I f p = 1 t h e n (5.7) follows f r o m L e m m a 5.1, 1 ~ (A m ~

An~2,

n ~ m < n ' ) . W e s u p p o s e t h a t (5.7) is t r u e for p < p ' . L e t p">~ 1 b e t h e l a r g e s t n u m b e r

~<p' such t h a t A~p,, satisfies t h e Q-condition. T h e n we h a v e Anp,, >~QV'-V"An~, a n d t h u s #n,,, >~ (P - P + 1)CA n. U s i n g t h e i n d u c t i o n h y p o t h e s i s a n d ~v=lt~n, ~,=1 t~,, , ,, 2 ~ ' _ >~xv,,-1. + /~,p,, we d e d u c e (5.7) for c h a i n s of l e n g t h p ' .

S e c o n d l y s u p p o s e t h a t we h a v e a c h a i n n ~< n~ < n2 < . . . < n p < n ' such t h a t no one of t h e w i d t h s An~ , 1 ~< v ~<p, satisfies t h e Q-condition. T h e n we h a v e

p ~< C l o g [ l o g An [ (5.8)

since in t h i s case i t follows t h a t An, > ~ ~ _{n p} ~> ! t)v-1A N o w a c c o r d i n g to L e m m a _{2 ~ n .} 5.2, 1 ~ we h a v e

1 Q ' - I A n ~< C A . ]log Anl w h i c h p r o v e s (5.8).

N o w we p r o v e 2 ~ W r i t e N n = n ' - n a n d M n = C l o g l l o g An] (see (5.8)). W e m a y s u p p o s e t h a t M n / N n is small. L e t E~' be t h e c h a i n {vlv 6 E~, n ~ v < n'} a n d d e n o t e its l e n g t h b y N~', i = 1 , 2 . . . k - 1 . W e s u p p o s e t h a t N [ > M n , 1<~i<~], a n d

N[~Mn,

ARKIV F61~. MATRMATIK. Bd 7 nr 8 j < i < ~ k - 1 . I t follows f r o m (5.8) t h a t a t least one w i d t h Am has to satisfy the Q- condition where m is one of the M~ + 1 first n u m b e r s of E ; (i ~<j). A p p l y i n g (5.7) we o b t a i n

Z / ~ , ~> (N[ - Ms)

CA~.

(5.9)

v e E ~

Observing t h a t

we deduce f r o m (5.9) t h a t

J

~ N ; - i M , > ~ N , - ( k - 1 ) M ,

i = l

i >~ 2

v = n iffil veE~

On the other hand, we k n o w f r o m L e m m a 5.1, 2 ~ t h a t /x~ ~< C(A~ log A , ) 2.

T h u s /Y~ ~< C (10g A~) 2.

L e m m a 5.2 is n o w proved.

We n o w r e t u r n to the problem posed in t h e beginning of this section: Given 8 > 0 we let 8' = 8 3 (8 small). T h e n if A N < 8 ' for some index N it follows f r o m L e m m a 5.2, 1 ~ t h a t A n < 8, n/>~T Hence this 2V would be a solution of o u r problem. To find -1A where such a n index we successively d e t e r m i n e indices np such t h a t An~+l < 2 n,, nv+ 1 is t h e smallest index > n v w i t h this p r o p e r t y , a n d n o = 0 . Suppose t h a t A,~ >~ 8', v ~<p a n d t h a t A~p+x < 8'. A p p l y i n g L e m m a 5.2, 2 ~ we o b t a i n

P p P

N = nv +1 = ~ (n~+l - n,) ~< C Z (log A,~) ~ ~< C ~ (log (8' 2")) ~.

v=O 'v=O ~,=0

B u t obviously p ~< C [log e' [ a n d inserting 8' = e ~ we h a v e N ~< C i (log (8'2~)) 3 ~< C(log e-1) a.

V = 0

*

Of course the same estimate is true if t h e parallel slit d o m a i n is 0-angled. L e t To(Z) be the function of T h e o r e m 3.1. T h e following t h e o r e m follows at once f r o m the a b o v e (for Gr6tzsch's m e t h o d , see p. 115).

T h e o r e m 5.1. Let qvo(z)=limn~or zn(z), where the z~(z) are determined by GrStzsch's method. Then there exist constants C and q, 0 < q < l, independent o / n such that

]~o(z)-zn(z)[ <r

in a n y / i x e d closed subset o / D .

Remark 1. According to T h e o r e m 3.1, t h e modified iterative process shows t h a t all widths are < 8 after N~ = 0 (log s -1) m a p p i n g s . Since t h e n u m b e r 2Y~ here means

I. LIND, Iterative methods in conformal mapping

a number of mappings of ( k - 1)-connected domains this estimate and the one above concerning GrStzsch's process are not comparable from a practical point of view (k ~> 3). However, the modified process can be approximated as follows. For example, if k = 3 we first perform N~ mappings of simply connected domains involving slits which are numbered 1 and 2. If N~ is large then these slits are "nearly" rectilinear.

Then we perform N2 mappings involving slits which are numbered for example 2 and 3 so that they become "nearly" rectilinear etc. Given s > 0 we m a y seek a num- ber Nk(e) such that all widths are < s after Nt,(e) mappings (in the indicated mariner) of simply connected domains. We indicate the proof of the following estimate:

~ k ( ~ ) = 0 ( ( l o g ~:--l)k--1). (5.10)

This estimate is true for k = 2 (the proof of Theorem 3.1) and we suppose t h a t it is true for k - 1 . Let D be a given k-eoimeeted domain and let/(z) EE'(D), ~ =/(D).

B y excluding one arbitrarily chosen b o u n d a r y component of 0 ~ we obtain a ( k - 1)- connected domain ~ ' . Consider the set of all such domains g2' obtained under all mappings/(z) E E'(D). From the induction hypothesis and the compactness of E'(D) it is obvious that it is possible to choose a number N =Nk_l(e 2) = 0 ((log e-l) ~-2) such t h a t the approximate iterative process described above gives slits of widths < s 2 after N mappings starting from a n y one domain ~ ' .

We write the approximate mapping of D as o~ =/~(/~_l(...(z)...))= Fn(z) where the /~ refer to mappings of ( k - 1 ) - c o n n e c t e d domains and where these mappings are composed of N mappings of simply connected domains. L e t D ~ - F~(D) and let A, be the maximal width of the slits constituting 0D~. From the compactness of E'(D) and the exponential rate of convergence of the modified process it follows t h a t if A ~ > e for v~<n then A~<~Cqn, 0 < q < l (e is supposed to be small). Thus necessarily n ~< C log e -1 and Nk(e ) = n_Nk_l(s ~) = 0 ((log e-l) k ').

Remark 2. We conclude this section with a description of an iterative technique which is intermediate between GrStsch's process and the modified iterative process.

Suppose for example that we want to map the k-connected domain D coo_formally onto a zero-angled parallel slit domain. Using induction we suppose t h a t this map- ping is possible for a n y domain of connectivity ~ < k - 1 and that these mappings have the extremal property of Corollary 4.1. W i t h the notation of Section 2 we let z~+l(zn) = z~ + al /zn +... E F,'(D(, ~)) (~) where D(, ~) is a domain of connectivity k~, 1 ~< k~ ~ k - 1, bounded b y k~ of the continua C(1 n), C(~ ~) ... C(k n). The choices of these can be made arbitrarily with the restriction t h a t each index 1, 2, ..., k must occur infinitely often. Let the width of C~ (~) be Aw, v = l , 2 ... k and let A~=Max~ Avn.

Suppose that lim~_+oo An>0. Then there exists a sequence {nt} such that D(, ~) is bounded b y C (n~) m~ ' Vm, ' ' " ' C(~) C (~) (1 ~<p ~< k - 1 ) , (where the indices are fixed), and such m p '

that the maximal width of these continua is at least A > 0. Then there exists A ' > 0

a(nO -->

such that Re a(1 n~) ~> A' because otherwise Re 1 0 , D(, ~'0-~ D , (at least for a subse- quence which we suppose already chosen). Thus there exists ](~) = ~ +al/~ + ... E Y~'(D,) mapping D,, which is not a parallel slit domain, onto a zero-angled parallel slit domain with Re a 1 =0, which contradicts the assumption t h a t Corollary 4.1 is true.

We have zn (z) = z + ,=0 + . . .

Z