1549055259-Ubiquitous_Quasidisk__The__Gehring_

8.1. QUASIDISKS AND HYPERBOLIC SEGMENTS 103

:oo

H

· W

g

f

h

0 0

FIGURE 8.4

and we obtain (8.1.6) with c = c 2.
Finally, since Dis bounded, we can find a K^2 -quasiconformal self-mapping f of

R
2
which maps D conformally onto B with f(oo) = oo. Fix z E /,choose zo E aD
so that

lz - zol = dist(z, 8D),

and let w 1 , w2, w, and wo be the images of z1, z2, z, and zo under f.
Since f (T) is a hyperbolic line in B , it is easy to check that

min lw - Wjl :::; 2 dist(w, BB ) :::; 2 lw - wol

J=l,2

and hence

min lzj - z l :::; 2c~ lz - zol = 2c~ dist(z, aD)

J=l,2

by Corollary 1.3. 7, where c 3 = e^8 K2. If /j is the component of I\ { z} which has
Zj as an endpoint, then

and we obtain (8.1.7) with c = 2 c2 c5. D

B

z I f

W2

Zo

W1

FIGURE 8.5