1549055259-Ubiquitous_Quasidisk__The__Gehring_

140 10. THIRD SERIES OF IMPLICATIONS

for j = 1, 2 by (5.1.3) in the proof of Lemma 5.1.2. Thus

hD(z1, z2) = lu(z1) - u(z2)I

:S iu(z1)-uB,I + iuB, -uB 2 I + luB 2 -u(z2)I

:S bfo(z1,z2) +b+4

and we obtain (10.2.4) with c = 2 b + 4 from Lemma 10 .2.1.

10.3. Hyperbolic metric and hyperbolic segments

D

We begin with the following lower bound for the hyperbolic distance hD(z 1 , z2)
similar to that given in Lemma 3.3.5.

LEMMA 10.3.l (Gehring-Palka [68]). If D is simply connected, then

I

dist(z1, 8D) I
(10.3.2) log dist(z
2
, oD) :::; 2 hD(z^1 , z2)

for z1, z 2 ED.

PROOF. As in the proof of Lemma 3.3.5 we see that

(^1) (
dist(z2,8D))
1
og. :::; og (l---z1-z2l+dist(.-'--------'-z1,8D))
d1st(z1, 8D) d1st(z 1 , 8D)
=log ( di~:(z~,~b) + 1) :::; 2 hD(z^1 , z^2 ).
Interchanging z 1 and z 2 yields
(
dist(z 1 , 8D))
log dist(z2,8D) :::; 2hD(z1,z2)
and we obtain (10.3.2). D
We complete the first chain of implications in this chapter by establishing the
following result.
THEOREM 10.3.3 (Gehring-Osgood [67]). Suppose that D is simply connected
and that
(10.3.4)
for each z1, z2 E D where a is a constant. Then for each hyperbolic segment I
joining z 1 , z2 ED and each z E /,
length(!) :Sc lz1 - z2I,
(10.3.5) min length(l
J=l, 2^1 ):::; c dist(z, 8D),
where /1, 12 are the components of/ \ { z} and c = c( a).
PROOF. Fix z1, z2 E D , let / be the hyperbolic segment joining z 1 to z 2 in D ,
and set
(10.3.6) r =min (sup dist(z, 8D), 2 lz1 - z2 I).
z E"f