1549055259-Ubiquitous_Quasidisk__The__Gehring_

108 8. FIRST SERIES OF IMPLICATIONS

Next let /3 1 ,/3 2 denote the components of 8D \ (a 1 U a2), labeled so that /3j c 'Yj

and choose w 1 E /3i and w2 E f32. Then

lwi - z1I:::; diam('Yi):::; blz1 - z2I = br.

If 81 , 02 denote the components of 8 D \ { w1, w2}, then

s = min diam(aj):::; min diam(oj):::; blw1 - w2I

J=l,2 J=l,2

by (8.5.2) and hence

s s

lw2 - zil 2'. lw2 -w1l - lw1 - zil 2'. b - br 2'. 2 b.

Thus for br < t < s /2b each circle { z : lz - z 1 I = t} intersects a1 and a2 and
separates /3 1 from /3 2. Hence each such circle contains an arc "( which joins a1 and
a2 in D , i.e., an arc 'YE r. Lemma 1.3.2 implies that
1 s
1 = mod(f) ;:::: 2 7r log 2 b 2 r

from which (8 .5.5) follows.

Finally, by (8.5.5) and Lemma 1.3.3,

(

mod(f*):::; 7r r s + 1 )2 :::; 7r (2b^2 e^2 7r+1)^2

as desired. 0

8.6. Quadrilateral inequality and quasidisks

Suppose that Di and D2 are Jordan domains in R

2

and that f : Di -7D2 is

a K-quasiconformal mapping. Then by Theorem 1.3.11, f has a homeomorphic

extension which maps D 1 onto D 2. We begin here by studying the boundary

correspondence¢: 8 D 1 -70D2 induced by f.

Suppose 9) : Dj-7H is conformal for j = 1, 2. Then 9) has a homeomorphic

extension to D j,

h = 92 ° f^0 91

1

is a self-homeomorphism of H which is K-quasiconformal in H , and
¢ = 92i^0 '!/;^0 91
where'!/; is the boundary correspondence induced by h. Hence in order to study the
mapping ¢ modulo conformal mappings, it is sufficient to consider the case where
Di= D2 =Hand where ¢(00) = oo.
We have next an important characterization due to Beurling and Ahlfors [23]
for the boundary correspondences ¢ : 8H-78H induced by quasiconformal self-
mappings f of H. We then indicate why each such mapping ¢ is the boundary
correspondence for a quasiconformal mapping f which is , in addition, bilipschitz
with respect to the hyperbolic metric (Ahlfors [7]).
LEMMA 8.6.1. Suppose that¢: Ri-7Ri is the boundary correspondence induced
by a K -quasiconformal mapping f : H-7 H. Then ¢ is a homeomorphism and
~ < ¢(x + t) - ¢(x) < k
(8.6.2) k-¢x ( ) -<Px-t ( ) -

for all real x and t > 0, where k is a constant which depends only on K.