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3.3. BOUNDS FOR HYPERBOLIC DISTANCE 35

where the infimum is taken over all rectifiable curves (3 which join z 1 and z 2 in D.

Again there is a unique hyperbolic segment 'Y in D for which

hn(z1, z2) = j Pn(z) ldzl.

Then

hn(z1, z2) = hB(g(z1), g(z2))

and g preserves the class of hyperbolic segments in D and B.

Finally from the Schwarz lemma and t he Koebe distortion t heorem (Ahlfors

[8]), it follows that

(3.2.1)

1 2

2 dist(z, BD) ::::; Pn(z) ::::; dist(z, BD )

for z E D , where dist(z, BD) denotes the Euclidean distance from z to BD. In

particular,

1 1

Pn(z ) = - - = ---

Im(z ) dist(z, BD)
if D is the upper half-plane H and

(3.2.2) p D ( r e iO ) = -7r sec (-7r(}) -^1

a a r

if D = S(a).
We show in what follows how a quasidisk D can b e characterized in four different
ways by comparing the Euclidean and hyperbolic geometries in D and its exterior
D *:

1° Bound for hyperbolic distance. This asserts that the hyperbolic distance

between points in D is bounded above by a function of the ratio of t he

Euclidean distance between the points and t heir E uclidean distances from

BD.

2° Geometry of hyperb olic segments. This describes t he Euclidean length

and posit ion of hyperbolic segments in D in terms of their endpoints.

3° Min-max property of hyperbolic segments. This assumes t hat, up to a

constant factor, the endpoints of hyperbolic segments 'Y in D minimize

and m aximize the Euclidea n distance b etween points in 'Y and in D *.

4° Hyperbolic reflection. This asserts the existence of a reflection in BD that

is bilipschitz with respect to the hyperbolic distances in D and D *.

3.3. Bounds for hyperbolic distance
The hyperbolic distance hn is, by its nature, difficult to calculate in all but the
simplest of domains. However the following two metrics yield lower bounds for hn
when D is simply connected. The first is the Apollonian metric

(

lz1 - w1llz2 - w2I)

(3 .3.1) an(z1, z2) = sup log I II I ,

W1,w2E8D Z2 - W1 z1 - W2

see Beardon [20] and Gehring-Hag [58], and t he second is the distance-ratio metric

(3.3.2) Jn(zi,. z (^2) ) =log ( dist(zlz1- z2I ) ( lz1-z2I )
1 , BD) +
(^1) dist(z2, BD) + (^1) '
see Gehring-Palka [68]. Moreover , these metrics also furnish upper bounds for hn
whenever D is a qu asidisk.