1549055259-Ubiquitous_Quasidisk__The__Gehring_

where

2.2. THE THREE-POINT CONDITION

a = { c

1

sc(a) if 0 <a:::; 7r/2,

if 7r / 2 < a < 7r.

23

The following counterpart of inequality (2.2.1) yields another characterization

for the class of quasidisks.

DEFINITION 2 .2.2. A Jordan domain D satisfies the three-point condition if

there exists a constant a 2:: 1 such t hat for each pair of points z 1 , z 2 E [) D \ { oo},

(2 .2.3 ) min diam('yj) :::; a iz1 - z2 I

J=l,2

where / 1, / 2 are the components of 8D \ { z 1 , z 2 }.

This condition derives its na me from the fact that it implies t hat

(2.2.4)

for each point z in the component of [JD\ {z 1 , z 2 } with minimum diameter. Con-

versely (2.2.4) implies (2.2.3) with 2a.

Ahlfors' well-known three-point co ndition of a quasidisk is as follows.

THEOREM 2.2.5 (Ahlfors [5]). A Jordan domain D is a K -quasidisk if and only

if it satisfies the three-point condition with constant a , where K and a depend only

on each other.

We see from t he three-point condit ion that a quasidisk cannot have a ny inward

or outward cusps.

REMARK 2.2.6. The fact that all quasidisks satisfy the three-point condition

follows directly from Theorem 1. 3.4.

To see this, let D be a K-quasidisk. Then t here is a K -quasiconformal mapping

of R

2

onto itself with D = f(G), where G is a disk or a half-plane. If f(oo) =!= oo,

there is a Mobius transformation ¢with ¢(00) = f-^1 (00). Then F = f o ¢is a

K-quasiconformal m apping with F(¢-^1 (G)) = D and F(oo) = oo, and D is the

image of a disk or h alf-pla ne under F.

Next choose z 1 , z 2 =/= oo in 8 D and let (j = p-l ( Zj) E 8G. Let /^1 be the subarc

of 8G \ { ( 1 , ( 2 } of smallest Euclidean diameter and let / = F ( /^1 ) b e its image in

aD. Then for each ( E /^1 we must have t hat

i( - (11:::; 1(1-(2 1

b ecause G is a disk or a half-plane. Then Theorem 1. 3.4 yields

IF(() - F((1)I :S e^8 J(IF((1) - F((2)J,

or
iz - z1l :S e^8 J(lz1 - z2J,

for each z E / , which means that diam('y) :::; 2e^8 J(Jz 1 - z2J. Thus D satisfies the
three-point co ndition.
A different route to this implication will be presented in Chapter 8, where we
will also prove the sufficiency part of Theorem 2.2.5.

The criterion in Theo rem 2.2.5 is particula rly useful when proving that a given

domain is a quasidisk since the following res ult shows it is only necessary to ve rify

inequality (2.2.3) for relatively few pairs of points z 1 , z 2 E 8D.