1549055259-Ubiquitous_Quasidisk__The__Gehring_

10 .4. APOLLONIAN METRIC IN A QUASIDISK 147

by (10.4.15). Hence m < r/2. Next

C = 8D C {z: r - 1 ::=::: lz l ::=::: r + l}.

and by performing a preliminary rotation, we m ay assume that

{arg(z): z E R^2 } = [-¢,¢]
where¢> 0.
We shall show that ¢ ::::: 7f /6. For t his fix 0 < B < min(¢, 7f /2) and choose
w 1 , w2, w3, w4 E C so that arg(w2) = B, arg(w4) = -B and so that w1 and W3 are
positive and separate w 2 , w 4 in C. Then w 1 , w 2 , W3, w 4 is an ordered quadruple of
points on the K-quasicircle C with

Hence

r sin(B) ::=:::min ( lw2 - r l, lw4 - rl)::::: m < r/2

by Corollary 10.4.12 and B < 7f /6. Thus ¢ ::::: 7f /6 and

CC {z: r - 1 ::=::: lz l ::=::: r + 1, I arg(z)I ::=::: 7r/6}.

Let U denote the smallest closed disk with center on { z : lz l = r} which contains
C; by performing a second preliminary rotation we may assume that

U={z: lz-rl:=::b}.

Then b ::=::: m, for if b > 1, there exists as above an ordered quadruple of points
w1, w2, w3, W4 on C such that

max(lw1 - rl, lw3 - rl)::::: 1 < b = lw2 - rl = lw4 - rl

and b::::: m by Corollary 10.4.12.
Finally, set
m
f(z ) = -.
r - z

Then BC f(D) and

(

hD(O, oo) = hf(D)(m/r, 0) ::=::: hB(m/r, 0) =log --r+m).

r-m

Hence if

g(t) =log G ~:) -m

2

log G ~ ~),

then g'(t) > 0 for r < t < oo and we obtain

m^2 aD(O, oo) - hD(O, oo) 2: lim (g(s) - g(r)) = j

00

g'(t) dt > 0.

s -tc:>O r

COROLLARY 10.4.17. If DC R^2 is a K-quasidisk, then

(10.4.18)

for z1, z2 ED where c(K)---+ 1 as K---+ 1.

D