1549055259-Ubiquitous_Quasidisk__The__Gehring_

10.4. APOLLONIAN METRIC IN A QUASIDISK 145

PROOF. Set

Then

f'(u) = K(cu)K-^1 g(u)

where

Hence

g(u):::; g(l):::; 0, J'(u):::; 0, f(u):::; f(l) = 0

and, with (10.4.2),

v:::; f(u) + (2KcK-l - l)uK:::; (2KcK-l - l)uK:::; (2c)2(K-l)UK

for u ;::: 1. This implies (10.4.3) when u ;::: 1. The same argument with u-^1 in place

of u yields (10.4.3) when 0 < u :::; 1. 0

THEOREM 10.4.4. If f is a K -quasiconformal self-mapping of R

2

and if D is

• 2
  a proper subdomain of R , then
  (10.4.5)
  for z1, z2 E D where
  (10.4.6) b(K) = 2(K - l)log32.
  PROOF. By performing preliminary Mobius transformations we may assume
  that D, D' C R^2. Fix z 1 , z 2 ED and choose w 1 , w 2 E 8D so that
  lf(z1) - f(wi)llf(z2) - f(w2)I
  af(D)(J(z1), f(z2)) =log lf(z1) - f(w2)llf(z2) - f(wi)I"

Then by Theorem 1.3.10,

where

Hence
v:::; 32 2 CK-l) max(uK, u-K),

whence
log v:::; Kl log ul + b(K) :::; KaD(z1, z2) + b(K)

by Lemma 10.4.1. Thus

af(D)(J(z1), f(z2)):::; KaD(z1, z2) + b(K).

0

COROLLARY 10.4.7. If D is a domain in R^2 and if there exists a K-quasi-

conformal self-mapping f of R^2 which maps D conformally onto a disk, then

(10.4.8) hD(z1, z2) :::; K aD(z1, z2) + b(K).

PROOF. By (3.3.8) and Theorem 10.4.4,

hD(z1, z2) ht(D)(J(z1), f(z2))

= af(D)(J(z1), f(z2)) :::; K aD(z1, z2) + b(K).

0