1549055259-Ubiquitous_Quasidisk__The__Gehring_

8 .1. QUASIDISKS AND HYPERBOLIC SEGMENTS

H

iyj

iy

f

'lYJ+l

0

FIGURE 8.2

by Corollary 1.3.7. This yields

y

log-^1 - :::; (k + 1) log2c2 = c3,

Yj+I

and with (8.1.4) we obtain

1Yo lf'(iy)I dy = f 1Yj lf'(iy)I dy

0 j=O YJ+I

Finally, if x E 8H, then

00

:::; 2c3lf(iyo)I L::C;-j+i = c4lf(iyo)I.

j=O

If ( i Yo) I :=:: c2 If ( i Yo) -f ( x) I

by Theorem 1.3.4 again and thus

lf(iyo)I:::; c2dist(f(iyo),8H ).

101

D

This completes the proof of (8.1.3) with c 1 = c2 c 4 , a constant which depends only

onK. 0

THEOREM 8.1.5 (Gehring-Osgood [67]). Suppose that 'Y is a hyperbolic segment

joining z 1 and z 2 in a K-quasidisk D C R^2. Then there exists a constant c = c(K)

such that for each z E 'Y,

(8.1.6)

(8.1.7)

length("():::; clz1 - z2I,

min length('Yj) :::; c dist(z, 8D),

J=l,2

where 'YI, 12 are the components of 'Y \ { z}.

PROOF. By Lemma 8.1.1 there exists a K^2 -quasiconformal self-mapping f of

R
2
which maps D conformally onto B. By employing an auxiliary Mobius trans-
formation of the disk we may assume that f (z 1 ) and f (z 2 ) are real. Next let B' be
the open disk in B with f (z 1 ) and f (z 2 ) as diametral points. Then D' = 1-^1 (B')
is a bounded K^2 -quasidisk and 'Y is a hyperbolic line in D'. Since

dist(z, 8D') :::; dist(z, 8D)