1549055259-Ubiquitous_Quasidisk__The__Gehring_

3.8. HARMONIC QUASISYMMETRY 45

for j = 1, 2, 3 and we obtain

1 ¢

w(l,11;D) = -(arctana~ - arctanaf) = -,

n n

1 ¢

w(l,12;D) = -(arctana~ - arctana~) = -

n n

from (3.7.2) and the conformal invariance of harmonic measure. A similar argument

yields

1

w(-l, 11; D*) = -(arctan a~ - arctan ai),

n

w(-l,12;D*) = ~(arctanaj-arctana~)

n

and we conclude that

where

(3.8.6)

w(-l,12;D*)

w(-l,11;D*)

g(e, t) = arctan(tant e)

9 (e + ¢, t) - 9 (e, t)

9 (e, t) - 9 (e - ¢, t)

and t= <]_= __ a_.

p 2n-a

A technical but elementary argument then shows that

1 9 (e+¢,t)- 9 (e,t)

• < < 1
  c - 9 (e,t)- 9 (e-¢,t) -

for 0 :::; e -¢ < e + ¢ :::; n /2 and this yields the desired conclusion if /1 and /2

lie on the same ray of 8D. The general conclusion can now be deduced from this

special case.

The preceding example suggests the following quasisymmetry property. See

Krzyz [105].

DEFINITION 3.8.7. A Jordan domain Dis quasisymmetric if there exist points
z 0 E D , z 0 E D* and a constant c 2: 1 such that if 11 , 12 are adjacent arcs in 8D
with

then

THEOREM 3.8.8 (Krzyz [105]). A Jordan domain D is a K -quasidisk if and

only if it is quasisymmetric with constant c, where K and c depend only on each

other.

This result will follow from the discussion in Section 3.12 where it is proved that
a domain is quasisymmetric if and only if it satisfies the conjugate quadrilateral
inequality. See Definition 3.10.8 below.