1549055259-Ubiquitous_Quasidisk__The__Gehring_

42 3. CONFORMAL INVARIANTS

Then with (3.6.4) and (3 .6.5) we obtain

and

(1 + lf(w)l^2 ) (1 + lf(oo)l2) ~ (rilf(oo)l^2 + r~)lf(w) - 112

~ 2 M^2 (1 + lf(w)l^2 ) (1 + lf(oo)l2)

(1 + lf(w)l2) (1 + lf(oo)l^2 ) m^2 ~ (rflf(oo)l^2 + r~)lf(w) - 112

~ 2 (1 + lf(w)l2) (1 + lf(oo)i2)

from which (3.6.3) follows.
Finally to see that the inequalities in (3.6.2) are sharp for the case where D is
a disk, one need only examine the case where the geodesic / tends to the diameter
of Dor to a point of 8D.

Example 3.6.l leads to a different way of characterizing quasidisks in terms
their hyperbolic geodesics.
DEFINITION 3.6.6. A simply connected domain D has the geodesic min-max
property if there exists a constant c :'.:: 1 such that for each hyperbolic segment /
joining z1,z2 ED,

~ min lzj - wl ~ lz - wl ~ c max lzj - wl

c J=l,2 J=l,2

for each z E / and w ¢: D.

THEOREM 3.6.7 (Gehring-Hag [55]). A simply connected domain D is a K-

quasidisk if and only if it has the geodesic min-max property with constant c, where

K and c depend only on each other.

3. 
   
   
   
   7. Harmonic measure
      The harmonic measure of an open or closed arc 'Y c 8B at a point z E B is
      defined as
   
   
   
   

(3.7.1) w(z) = w(z, 1; B) =! P(z, () ld(I

where P(z, () is the Poisson kernel

1 1 - lzl^2

P(z, () = 27r lz - (12.

Thus w(z) is the unique function which is bounded and harmonic in B with bound-

ary values 1 and 0 at interior points of 'Y and 8B \ ry, respectively. Note that

w(O) = w(O, 1; B) = length(/).

27r

It easily follows that for every conformal self-mapping f of B we have that

w(z, "(; B) = w(f(z), f(r); B).

Suppose next that D C R

2

is a Jordan domain and that g is a conformal

mapping of D onto B. Then g has an extension which maps D homeomorphically

onto B. We define the harmonic measure of a n arc 'Y C 8D at a point z ED by

w(z) = w(z, 1; D) = w(g(z), g(r); B).

By the conformal invariance w(z) is independent of the choice of the mapping g.