1549055259-Ubiquitous_Quasidisk__The__Gehring_

11.3. HOMOGENEITY AND QUASIDISKS

since by monotonicity

But by (11.1.6)

and

so that

This means that

w(z, f)D,* n &G; G) 2: w(z, 1; D*).

w(z,S;D*) _::; ~

1T d(z, S)

4

1 - r:iT::C\ :::; w(z, &D* n &G; G) _::; c < 1

7Ty d(z, S)

lzi -zJ - 1 :S: d(z, S) _::; ( 7r(l ~ c))

2

+ 1.

153

0

11.2. Quasidisks and quasiconformal extension domains

THEOREM 11.2.1. If D is a K-quasidisk, then any K' -quasiconformal self-

mapping f of D has a cK' -quasiconformal extension as a self-mapping ofR

2

, where

c only depends on K.

PROOF. We may assume that D = g(B) for a K-quasiconformal mapping g of

R

2

. Then the mapping h = g-^1 of o g is a K^2 K'-quasiconformal self-mapping of
B and thus h as a K^2 K'-quasiconformal extension to R
2
by Example 5.3.2. Then
the mapping go h o g-^1 is a K^4 K'-quasiconformal extension of g, so we may take
c = K^4. D

Thus quasidisks are quasiconformal extension domains.

Next we prove that quasiconformal extension domains are homogeneous with

respect to the family of mapping QC.

THEOREM 11.2.2. If a simply connected domain D is a quasiconformal exten-

sion domain with constant c, then it is homogeneous with respect to QC ( c).

PROOF. For this, fix z 1 , z2 ED, let f be a co nformal mapping of D onto B , and
let Wj = f(zj ) for j = 1, 2. Then there exists a Mobius transformation g: B-+ B
such that g(wi) = w2 and hen ce h = 1-^1 o go f is a conformal se lf-mapping of D
which maps z 1 onto z2. Then since Dis a quasiconformal extension domain, h has

a c-quasiconform al extension to R

2

, and t hus D is homogeneous. 0

11.3. Homogeneity and quasidisks
We will give a proof due to Sarvas [153] for the case when the homogeneity
is with respect to quasiconformal maps of the domain. The argument is a clever
application of the three-point condition for quasidisks. See Theorem 2.2.5.

THEOREM 11.3.1. A Jordan domain D is a quasidisk if it is homogeneous with

respect to the family QC(K) for some fixed K.