1549055259-Ubiquitous_Quasidisk__The__Gehring_

11.4. EXTREMAL DISTANCE DOMAINS

PROOF. Choose z 1 E C1 and z2 E C2 so that

lz1 - z2I = dist(C1, C2).

Then by hypothesis there exists a point Wj E cj such that

lz1 - w1I ~ ~ diam(Cj) ~ ~ dist(C1, C2) = ~ lz1 - z2I

for j = 1, 2. By relabeling if necessary we may also assume that

lz1 - w1I :::; lz2 - w2I·

Let f : R

2

---+ R

2

be a Mobius transformation for which j(w2) = oo. Then

lf(z1) - f(z2)I = lz1 - z2IJw1 - w2I < ~ lw1 - w2I

lf(z1) - f(wi)I lz1 - w1llz2 - w2I - a lz2 - w2I

<~lz 1 -w1l+lz1-z2l+lz2-w2I < 4 a+l =t

-a lz2-w2I ~ a^2 '

and it is not difficult to show that

(^1) og (^4) v ri t = (^1) og (8Ja+l) < --2a+l.
a a
We conclude that
7r na
mod(f) = mod(f(r)) ~ Vt~ -
2

• log4 t a+ 1
  by Lemma 11.4.2.

159

0

A set E c R

2

is said to be a-convex where 1 :::; a < oo if each pair of points

z 1 , z 2 E E \ { oo} can be joined by a curve / C E such that

length(!) :::; alz1 - z2I·

If E C R^2 , then E is 1-convex if and only if E is convex in the usual sense.
We show next that domains with the extremal distance property are a-convex
for some constant a.

LEMMA 11.4.4 (Gehring-Martio [65]). If D C R

2

has the extremal distance

property with constant c, then D is a-convex where

(11.4.5) a= (5/3) exp(48c).

PROOF. Fix z1, z2 E R^2 , let r = I z1 - z2 I, and let C be a curve joining z1 and

z 2 in D. Next for j = 1, 2 let C1 denote the component of

CnB(z 1 ,r/4)

that contains Zj and let r D and r denote the families of curves joining C1 and C2

in D and R
2
, respectively. Then

mm. d' iam (C) r dist(C1, C2)

J=l,2^1 ~ - 4 ~ --'------'- 4

and hence by Lemma 11.4.3 and (3.11.4)

(11.4.6) mod(fD) ~ mod(f) ~ bo,

c c

where bo = 7r /6.