1549055259-Ubiquitous_Quasidisk__The__Gehring_

3.12. QUADRILATERALS AND HARMONIC QUASISYMMETRY 55

by (3.12.7), and by interchanging the pairs of indices (1, 2) and (3, 4) and the pairs

(2, 3) and (1, 4) when appropriate, we need only consider the following two cases:

1° lz3 - z4I:::; lz2 - z3I, lz4 - z1I,

2° lz4 - z1I:::; lz1 - z2I:::; !zJ -z4I:::; lz2 - z3I.

In the first case we see that

l¢(z2) - ¢(z4)I :::; c l¢(z2) - ¢(z3)I

by (3. 12 .13) applied to the triples of points {z 1 ,z4,z3}, {z2,z 3 ,z 4 }. This, in turn,

yields the upper bound in (3.12.11) with a= (b+ 1)^2.

In the second case we obtain

1¢(zi) - ¢(z3)I:::; c l¢(z2) - ¢(z3)I

from (3.12.13) applied to the triple {z3, z 2 , z 1 }. Next we choose z 0 on the arc of aB

with endpoints z 1 , z 2 so that

lzo - z1I = lz1 - z4I > (v'2 - l)lz1 - z2I,

by Remark 3.12.8. Then a numerical computation shows that lz2 - zol < lzo - z4I

and we obtain

l¢(z2) - ¢(z4)I:::; cl¢(z4) - ¢(zo)I:::; c^2 l¢(z4) - ¢(z1)!.

This pair of bounds yields the upper bound in (3.12.11) with a= (b + 1)^3.

To obtain the lower bound in (3.12.11), we observe that [z 1 , z 2 , z 3 , z 4 ] = 2

implies [zi,z4,z3,z2] = 2 by (3.12.7). Likewise [¢(z1),¢(z2),¢(z3),¢(z4)]-^1 = 1-

[¢(z1), ¢(z 4 ), ¢(z 3 ), ¢(z 2 )J-^1 and the lower bound follows from the upper bound. 0

We now apply Lemmas 3.12.2 and 3.12.9 to obtain the main result of this

section.

THEOREM 3.12.14 (Gehring-Hag [59]). Suppose that D is a Jordan domain in

R

2

. Then D is quasisymmetric with respect to harmonic measure if and only if D
satisfies the conjugate quadrilateral inequality.

PROOF. Suppose that D satisfies the conjugate quadrilateral inequality and

choose conformal mappings f : D -+ B and g : D* -+ B normalized so that ¢ fixes

the points i, -1, -i where

¢=h, h = go r^1 : aB -+ aB.

We shall show that D satisfies the harmonic quasisymmetry condition with respect

to the points w 0 = f-^1 (0) ED and w 0 = g-^1 (0) ED*.

For this suppose that Q = D(w 1 ,w 2 ,w 3 ,w 4 ) is a quadrilateral with mod(Q) =

1. Then by hypothesis, mod(Q) :::; c. The corresponding quadrilaterals f(Q) and
   g(Q) in B are B(z 1 , z2, z3, Z4) and B(h(z4), h(z3), h(z2), h(z1)) and (3.10.5) and
   (3.10.6) imply there exists a constant a= a(c) ~ 2 such that
   a - 1 < lh(z1) - h( z3)l lh(z2) - h(z4)I <a.
   (
   3
   ·
   12
   ·
   15
   ) a - lh(z1) - h(z4)! lh(z2) - h(z3)I -

Then Lemma 3.12.2 with ¢ = h implies that

~ < lh(z1) - h(z2)I < b

(^3 ·

12

·

15

) b - lh(z2) - h(z3)I - '