Let w (z) E S. I1 M~ <_ M

(1)

A R K I V F O R M A T E M A T I K B a n d 2 n r 21

C o m m u n i c a t e d 26 N o v e m b e r 1952 b y F . CAI~LSON

A n o t e o n t h e c o n s t a n t o f K o e b e

B y BENGT J . ANDERSSON

Let S be the class of analytic functions w (z)= a 1 z + a2 z2+ "'" t h a t are schlicht in the unit circle 7: I z l < 1. The function w (z) maps 7 on an open and simply connected domain

Dw.

We define

1 Inf Iwl,

M w = , l - ~

Sup Iwl .

d w = l a i I w ~ D w ]allwED w

I t is wellknown t h a t d~ _>88 (Koebe's constant), this limit being the best possible for M~_< c~. Here we shall determine a stronger limit t h a t depends on Mw.

T h e o r e m .

Let w (z) E S. I1 M~ <_ M

(1) d ~ - > 2 M 2 [ 1 2 M I ( i - M ] "

I t is allowed to put

w'

(0) = al = 1. Let w o (z) = :r z + ~2 z~ + "'" be a function in S that maps 7 on the circle I w I < M, slit along the segment (dw, M) of the real positive axis. The inverse functions of w (z) and Wo (z)are z ( w ) a n d z 0 (w):

z' (0) = 1, zo (0) = ~;1. The harmonic functions

y J ( w ) = l o g l ~ z ~ l and ~ p o ( w ) = l o g l ~ ) l

are regular and _< 0 in Dw and Dw 0 respectively. Any circle I w ] = r, d~ < r < M contains at least one point w (~ Dw. Further, if w approaches a point

w'

on the boundary of Dw we get

lim ~ (w)<log = ~ . ( I w ' l )

and ~o (w) has non-negative derivatives along the inner normals of the segment (dw, M). Then all conditions are satisfied for applying a lemma of BEURLING (1) that solves the problem. From this lemma we get y~o(0)_>v2 (0)

28 4 ] 5

(2)

B. J. ANDERSSON, A note on the constant o f Koebe

and hence follows I:r The function

Wo(Z)

is calculated by elementary methods. We obtain

':h =4dw [l+dM] -2->- 1.

Now it is easy to write this inequality in the form ( 1 ) a n d the lemma is proved. The lower limit in (1) is attained by

w = w o (z)

and is therefore the best possible.

R E F E R E N C E

(1) BELTRLING, A , E t u d e s s u r u n e p r o b l S m e de m a j o r a t i o n , Th~se, U p p s a l a 1933, p. 44.

T r y c k t den 16 februari 1953

416

Uppsala 1953. Almqvist & Wiksells Roktryckeri AB