1549055259-Ubiquitous_Quasidisk__The__Gehring_

36 3. CONFORMAL INVARIANTS

LEMMA 3.3.3. The Apollonian metric aD is a Mobius invariant pseudo-metric
in D. It is a metric whenever DD is not a proper subset of a circle or line.

PROOF. Equation (3.3.1) implies that aD is a pseudo-metric which is invariant
with respect to l\lfobius transformations; aD is a metric if, in addition, aD(z 1, z2) > 0
whenever z 1 and z 2 are distinct points of D. By the Mobius invariance we may
assume that z 1 = 0 and z 2 = oo. In this case

aD(O, oo) = sup log ( 1

1

wil

1

) = 0

w1,w2EBD W2

if and only if oD lies in a circle about 0. 0

LEMMA 3.3.4. The distance-ratio metric JD is a similarity invariant metric in
D c R^2.

PROOF. Equation (3.3.2) implies that JD is invariant with respect to similarity
transformations. To complete the proof, it suffices to show that

l = fo(z1, z3) :S fo(z1, z2) + fo(z2, z3) = r

for Z1, z2, Z3 ED. Let di = dist(zi, oD) for i = 1, 2, 3. Then from the inequa lities

lz1 - z2I + d2 > lz1 - z2I + lz2 - z3I + d3 > lz1 - z3I + d3

d2 lz2 - z3I + d3 - lz2 - z3I + d3

and
h -z3I + d2 > lz2 - z3I + lz1 - z2I + d1 > lz1 - z3I + d1
d2 - lz1-z2l+d1 lz1- z2l+d1
we obtain

()

lz1-z2l+d1 lz1-z2l+d2 lz2-z3l+d2

exp r =

d1 d2 d2

lz2 - z3I + d3

d3

> "------'----lz1-z2l+d1 lz1-z3l+d^3 lz1-z3l+d1

di lz2-z3l+d 3 lz1-z2l+d 1

lz1-z3l+d1 lz1-z3l+d3 _ (l)

d1 d3 - exp ·

lz2 - z3I + d 3

d3

0

The following two results indicate how aD and JD are related to the hyperbolic
metric hD in a simply connected domain.

LEMMA 3.3.5 (Gehring-Palka [68]). If D is simply connected, then

(3.3.6) aD(z1, z2) :S fo(z1, z2) :S 4 hD(z1, z2)

for z1, z2 ED.

PROOF. Fix z 1 , z 2 in D and w 1 , w 2 E 8D. Then

lz1 - Wjl < -------'--'---------'-'-lz1 - z2I + lz2 - Wjl < lz1 - z2I + 1

h -Wjl ~ lz2 -wjl - dist(z2,8D)

for J = 1, 2, whence

-----lz1-w1llz2-w2I - < (. lz1-z2I +l )(. lz1-z2I +l )

lz2 - w1llz1 - w2I - d1st(z1, oD) d1st(z2, oD)

and the first part of (3.3.6) follows from (3.3 .1) and (3.3.2).