1549055259-Ubiquitous_Quasidisk__The__Gehring_

144 10. THIRD SERIES OF IMPLICATIONS

by Lemma 10.3.1 and, as in (10.3.9),

length(r(w 1 , w 2 )):::; e 2b dist(w1,8D)^1 d. ( ds oD)

'Yi ist z,

:::; e^2 b dist(w1, oD) 1 2 PD(z) ds

'Yl

(10.3.18)

:::; 2be^2 bdist(w1,8D)

where 'Yi = 'Y(w 1 , w 2 ), and we again obtain (10.3.15).

We are now in a position to complete the proof of Theorem 10.3.3. By relabeling

and (10.3.11) we may assume that

dist(w1, oD) :::; dist(w2, oD) :::; 2 dist( W1, oD).

Then

j=l,2

(10.3.19)

j=l,2

:::; 4be^2 bdist(w2,8D):::; 4be^2 br:::; 4be^2 b lz1 - z2I

by (10.3.12) and (10.3.18). Next if z E ')',then either z E 'Y(Zj,Wj) and

(10.3.20) min length(r(zj,z)):::; length(r(zj,z)):::; bebl^2 dist(z,8D)

J=l,2

or z E 'Y(w 1 , w2) and

min length(r(zj, z)):::; length(r)/2:::; 2be^2 bdist(w 2 , oD)

J=l,2

:::; 2be^4 bdist(z, oD)

(10.3.21)

by (10.3.15).

c = 4be^4 b.

Hence (10.3.5) follows from (10.3.19), (10.3.20), and (10.3 .21) with

10.4. Apollonian metric in a quasidisk

If D is a simply connected domain, then

aD(z1,z2):::; 4hD(z1,z2)

D

for z 1 , z 2 E D by Lemma 3.3.5. We show here that this inequality can be reversed

if D is a K-quasidisk, namely that

hD(z1, z2):::; caD(z1, z2)

for z 1 , z 2 ED where c = c(K). We begin with the following result.

LEMMA 10.4.1. Suppose that K 2 1 and c 2 ~. If u > 0 and

(10.4.2) V + 1:::; CK-l(U + l)K,

then

(10.4.3)