1549055259-Ubiquitous_Quasidisk__The__Gehring_

56 3. CONFORMAL INVARIANTS

f

h

g

FIGURE 3.4

whenever lz 1 - z2I = lz2 - z3I, i.e., whenever z1,z2 and z2,z3 are the endpoints of

a pair of adjacent open arcs /l and 12 in 8B with

w(0,11;B) = w(0,12;B).

Now (3.12.16) with z 1 = -i, z 2 = 1, z 3 = i and the fact that h fixes the points

i,-1,-i imply that the distance between any pair of the points h(l), h(i), h(-1),

h( -i) is at least

2

2d= ~ < v'2.

vb^2 +1 -

Since length(Tj)::::; 7r, /j contains at most two points of {1, i, -1, -i}, h(Tj) contains

at most two points of {h(l),h(i),h(-1),h(-i)}, and

length(h(Tj))::::; 2(7r - d)

for j = 1, 2. From this.it follows that

1 < length(h(/1)) , length(h(/2)) < 7r - d = m,

• lh(z1) ~ h(z2)I lh(z2) - h(z3)I - sin(7r - d)
  where m = m(d). Thus

w(O, h(T 1 ); B) = length(h(/ 1 )) < mb

w(O, h(/2); B) length(h(/ 2 )) -

and we conclude that D is quasisymmetric with respect to harmonic measure.

Suppose next that D satisfies the harmonic quasisymmetry condition with re-

spect to the points wo E D and w 0 E D*. We introduce conformal mappings f and

g as before and let h be the induced homeomorphism

h =go f-^1 : 8B ---+ 8B.

Then because D satisfies the harmonic quasisymmetry condition, it is easy to see

that there exists a constant b such that (3.12.10) holds whenever lz 1 -z 21 = lz 2 -z3I.