1549055259-Ubiquitous_Quasidisk__The__Gehring_

10.3. HYPERBOLIC METRIC AND HYPERBOLIC SEGMENTS

We shall consider the cases where

(10.3.7)

(10.3.8)

separately.

r < max(dist(zj, 8D)),

J=l,2

r ~ max(dist(zj, 8D))

J=l,2

Case where r < supzEi dist(z, 8D)

If r < dist(z1, 8D), then r = 2 lz1 - z2I,

2 lz1 - z2I < dist(z1, 8D) :S 2 dist(z, 8D)

for z on the segment f3 = [z 1 , z 2 ] C D , and

f f 2 4lz1 - z2I

hD(z1,z2) :S J1/D(z)ds :S } 13 dist(z,8D) ds :S dist(z1,8D) :S 2.

Since

hD(z1, z) :S hD(z1, z2)

for z E -y, Lemma 10.3.1 implies that

for z E -y. Thus

(10.3.9)

and

(10.3.10)

e-^4 dist(z 1 , 8D) :S dist(z, 8D) :S e^4 dist(z 1 , 8D)

length("() :S e

4

dist(z^1 , 8D)! dist(~~ aD)

:S e^4 dist(z 1 ,8D)! 2pD(z)ds

= 2e^4 dist(z 1 , 8D) hD(z1, z2) :S 8e^4 lz1 - z2I

length('Y(z 1 ,z)) :S 8e^4 lz1 -z2I :S 4e^4 dist(z1,8D)

:S 4e^8 dist(z, 8D)

141

for z E -y. If r < dist(z 2 , 8D), then we obtain (10.3.9) and (10.3.10) by reversing
the roles of z 1 and z 2 in the above argument. Hence (10.3.7) implies (10.3.5) with
c = 4e^8.

Case where r 2:: supzEi dist(z, 8D)

In this case there exists z 0 E 'Y such that

dist(z 0 , 8D) =sup dist(z, 8D) ~ r.

zE"'f

For j = 1, 2 let mj be the largest integer for which

2mJ dist(zj,8D) :Sr

and let Wj be the first point of -y(zj, z 0 ) with

dist(wj, 8D) = 2mJ dist(zj, 8D)

as we· traverse 'Y from Zj towards zo. Then

(10.3.11) dist(wj, 8D) :Sr< 2 dist(wj, 8D).