44 3. CONFORMAL INVARIANTS
is sense-preserving with ¢(1) = l. We show first that ¢(z) = z.
For this let 11 and 12 be the upper and lower halves of 8B labeled so that
i E /'l · By the conformal invariance of harmonic measure,
w(zo, f (1'1); D) = w(O, ')'1; B) = 1/ 2 = w(O, ')'2,; B) = w(zo, f (1'2); D)
and hence by (3.8.2) and (3.8.3)
w(O, ¢('? 1 ); B) = w( oo, ¢(1'1); B *) = w(z 0 , go ¢(1'1); D*)
= w(z 0 , f(/'1); D*) = w(z 0 , f(/'2); D*)
= w(z 0 , go ¢(1'2); D*) = w(oo, ¢(1'2); B *)
= w(O, ¢('? 2 ); B).In particular we see that ¢(-1) = -1 and, since h is sense-preserving, we see that
¢(1'1) = 1'1 ·
Next let l'l and ')' 2 be the right and left halves of the upper half of 8B labeled
so that ei 7f /^4 E l'l · Then as above
w(zo, f(/'1); D) = w(O, ')'1; B) = 1/4 = w(O, ')'2; B ) = w(zo, f(/'2); D)and
w(O, ¢(1'1); B) = w(z 0 , f (1'1); D) = w(z 0 , f (1'2); D) = w(O, ¢(1'2); B).
Thus ¢( i) = i and ¢(1' 1 ) = ')' 1. Proceeding in this way we obtain
¢( e27fi t) = e27fi tfor all t E [O, l] of the form t = m 2-n where m, n E Z. Hence by continuity
¢( z) = z for z E 8B.
We conclude that f and g together define a self-homeomorphism h of R
2
whichis conformal in BUB* and hence in R
2
. Thus his a Mobius transformation and
D = h(B) is a disk or half-plane. 0
The following counterpart of Theorem 3.8. l for the case where D is a sector of
angle a suggests what the situation is for quasidjsks.
EXAMPLE 3.8.4. If D = S(a) and if ')' 1 , ')' 2 are adjacent arcs in 8D with
(3.8.5) w(l,')'1;D) = w(l,')'2;D),t hen
where
~w(-l,')'1;D):::; w(-l,')'2;D):::; cw(-l,')'1;D*)
c
7!'1-t
c=--2t - 1) t =min ( a , 27r-a).
27r - a a
We shall indicate how these bounds are obtained. Suppose that 0 < a :::; 7r and
that a 1 ei a./^2 , a 2 ei a./^2 , a 3 ei a./^2 are consecutive endpoints of adjacent arcs l'l and
1'2 in 8D for which (3.8.5) holds. Suppose next that 0 :::; a 1 < a 2 < a 3 and set
7r 7r ¢
p=-, q= , -=w(l,')'J;D)
a 27r-a 7r
for j = 1, 2. Then f(z ) = zP maps D conformally onto the right half-plane H so
that
f (1) = 1 and