1549055259-Ubiquitous_Quasidisk__The__Gehring_

148 10. THIRD SERIES OF IMPLICATIONS

PROOF. Fix z 1 , z 2 ED. Then

hn(z1, z2)::::; >..(K) an(z1, z2)

by Theorem 10.4.14 if

while
hn(z1, z2)::::; ( K^2 + >..(K)^1 l^2 b(K^2 )) an(z1, z2)
by Theorem 10.4.9 otherwise. Hence (10.4.18) holds with

c(K) =max ( >..(K), K^2 + >..(K)^112 b(K^2 )).

Next >..(K) --+ 1 and b(K) --+ 0, whence c(K) --+ 1, as K --+ l.

10.5. Apollonian metric and hyperbolic segments

D

The following consequence of Theorem 10 .3.3 completes t he second chain of
implications in this chapter.
THEOREM 10.5.l. Suppose that D is simply connected and that
(10.5.2) hn(z1, z2)::::; ban(z1, z2)
for each z 1 , z 2 E D where b is a constant. Then for each hyperbolic segment 'Y
joining z 1 , z 2 E D and each z E 'Y,

length("!)::::; c lz1 - z2I,

(10.5.3) min length('Yj)::::; c dist(z, aD)

J=l,2

where 'Yi, "(2 are the components of 'Y \ { z} and c = c( b).

PROOF. Lemma 3.3.5 and (10.5.2) imply that

hn(z1,z2)::::; ban(z1,z2)::::; bfo(z1,z2)

for each z 1 , z2 E D. Hence (10.5.3) follows for each hyperbolic segment 'Y joining

z1, z2 E D and each z E "( from Theorem 10.3.3. D