3.3. BOUNDS FOR HYPERBOLIC DISTANCE 37Next let 'Y be the hyperbolic segment joining z 1 and z 2 in D. Then1 1.
----,---------,-----..,.--< < 2 p D ( Z)
dist(zj, 8D) + Jz - Z j J - dist(z, 8D) -for z E 'Y by (3.2.1) and thus
Adding these inequalities for j = 1, 2 yields the second part of (3.3.6). 0
EXAMPLE 3.3.7. If D is a disk or half-plane, then
(3.3.8)
for z 1 , z 2 ED. The second part of (3.3.8) holds with equality whenever Dis a disk
and z 1 , z 2 lie on a diameter and are separated by the center of D.
Since aD and hD are invariant with respect to Mi::ibius transformations, we may
assume that D = B and that z 1 = 0 and z 2 = r where 0 < r < 1 in the proof for
the first part of (3.3.8). Then
(
aB(O, r) = sup log I lw1 -rl) I =log (l+r) -- = hB(O, r).
lw1l=lw2l=l W2 - r 1 -rThe inequality in the second part of (3.3.8) follows from Lemma 3.3.5.
Next since aD and JD are invariant with respect to similarity mappings, we
may assume that D = B and that z 1 = r 1 < 0 and z2 = r2 > 0 for the proof of
equality in the second part (3 .3.8) in which case
The following counterpart of Example 3.3.7 for the case where D is an angular
sector suggests what happens when D is a quasidisk.
EXAMPLE 3.3.9. If D = S(a) where 0 < a:::; 7r, then
(3.3.10)
7f
for z 1 , z2 E D whenever lz1 I = Jz2 I or zif z2 is real. The bound - is sharp.
a
For this suppose that Z j = r ei^81 where -a/ 2 < 81 < 82 < a/2. Then
(3 .3.11)