1549055259-Ubiquitous_Quasidisk__The__Gehring_

3.6. MIN-MAX PROPERTY OF HYPERBOLIC SEGMENTS 41

by (3.5.7) and we obtain

min length(lj)::; a(c + 1) length ( a)::; b(c + 1) dist(z, 8D)

J=l,2

from (3.5.9) where b = ab 1. D

Thus by Theorem 3.4.5, a domain D C R^2 is a quasidisk if and only if it is

simply connected and uniform.

The following result illustrates the connection between quasidisks and uniform

domains for the case where D is multiply connected.

THEOREM 3.5.10 (Gehring [49], [53], Gehring-Hag [55]). If D is a uniform

domain in R^2 , then there exists a K such that each component of R

2

\ D is either

a point or the closure of a K -quasidisk. The converse also holds if D is finitely

connected.

3.6. Min-max property of hyperbolic segments
A second way to characterize quasidisks in terms of the Euclidean properties
of their hyperbolic segments is suggested by the following less familiar property of
hyperbolic segments in a disk.

EXAMPLE 3.6.1 (Gehring-Hag [55]). If Dis a disk or h alf-plane and if I is a
hyperbolic segment joining z 1 , z 2 ED, then

(3 .6.2) __!._ min lz j - wl ::; lz - wl ::; v'2 max lzj - wl

J2 J=l,2 J=l ,2

for each z E / and w rf. D. The constant J2 cannot be replaced by a smaller
number in either of these inequalities.

For a proof of this fix z E /, w rf. D and let

r. - --=---'-lzj -wl

J - lz - w l

for j = 1, 2. We want to show that

(3.6.3) m = min rj ::; h,
J=l,2

1
M = max r j 2'. "'.
j=l,2 v 2
Choose a Mobius transformation f which maps z 1 , z , z2 onto 0 , 1 , oo, respec-
tively. Then

whence

(3.6.4)

lf(zj) - f(w)I IJ(oo) - l l

rj = lf(zj) - f(oo)I lf(w) - ll'

lf(w)llf(oo) - ll = r1lf(oo)llf(w) -11,

lf(oo) - ll = r2lf(w) - ll.

Since f(l) is a hyperbolic geodesic in f(D), f(w) and f(oo) lie in the left half-plane,

1 + lf(w)l^2 ::; lf(w) - 112 ::; 2 (1 + lf(w)l^2 ),

1 + lf(oo)i2::; lf(oo) - 112 ::; 2 (1 + lf(oo)i2).

(3.6.5)