150 11. FOURTH SERIES OF IMPLICATIONS11.1. Harmonic bending and quasidisks
We show first that a Jordan domain DC R^2 is a quasidisk if and only if it has
the harmonic bending property, that is, if each arc I C oD does not bend too close
to oD \/,the complement of I in oD. See Fernandez-Hamilton-Heinonen [41].
We begin with the following bound for the harmonic measure of a boundary
arc in a Jordan domain. See McMillan [128].
LEMMA 11.1.1. Suppose that D is a Jordan domain in R^2 and that/ is an arc
in oD. Then(11.1.2)^4 ( diam(!) )112
w(z 0 , 1; D) :::; - arctan d " ( )
7r ist zo, /
for each point z 0 E D.
PROOF. Suppose that zo ED, choose z1 E /so that
lzo - z1I = dist(zo,/),
and let d = diam(!). Then by a change of variable we may assume that d = 1,
zo = -lzol < 0, and z1 = 0. We may also assume that lzol > 1 since otherwise4 4 ( 1 )112
w(z 0 ,1;D):::; 1 =:; arctan(l):::;:; arctan Foland we obtain (11.1.2).
Since D is a Jordan domain with / c oD n B, there exists an arc /3 1 which
joins a point w1 E 8B to oo such that
(/31 \ {w1}) n (Du B) = 0.
Let D 1 be the domain with 8B U /3 1 as its boundary and let /l = oB \ { w 1 }. Then
w(z, /; D) -w(z, /1; D1) is harmonic in D \ B,
w(z,1;D) = 0:::; w(z,11;D1)
in o(D \ B), and hence
(11.1.3)
Next let /32 denote the part of the real axis joining w 2 = 1 to oo, let D 2 be the
domain with 8B U /32 as its boundary, and set /2 = oB \ { w 2 }. Then w(z, 12 ; D 2 )
is harmonic in D 2 and
(11.1.4)by the solution of the Carleman-Milloux Problem. See, for example, Nevanlinna
[138, pp. 107-113].
Finally, let
f(z) = ~ ( Vz + ~)
so that f(l) = 1, f(-1) = -1, and
f(zo) = ~ ( M -k) =~(cot(¢) - tan(¢))= tan~ 2 ¢)
where
(1 ) 1/2
¢> = arctan Fol