1549055259-Ubiquitous_Quasidisk__The__Gehring_

146 10. THIRD SERIES OF IMPLICATIONS

THEOREM 10.4.9. If DC R^2 is a K-quasidisk, then

(10.4.10) hD (z1, z2) :'S: K^2 aD (z1, z2) + b(K^2 )

for z1, z2 ED.

PROOF. By hypothesis, there exists a K-quasiconformal self-mapping f of R^2
which maps D onto a disk B. The existence theorem for the Beltrami equation
implies there exists a K-quasiconformal self-mapping g : B-+ B such that g o f is
conformal in D. See, for example, Ahlfors [6] or Lehto-Virtanen [117]. Reflection
in 8B extends g to a K-quasiconformal self-mapping of R^2. Then J = go f is
K^2 -quasiconformal and we can apply Corollary 10.4.7 to obtain (10.4.10). 0

The bound in (10.4.10) can be replaced by an inequality of the form

(10.4 .11) hD(z1, z2) :'S: c(K) aD(z1, z2)

where c(K) -+ 1 as K-+ l. To show this, we must first establish (10.4.11) for the
case where aD(z 1 , z 2 ) is small. The proof for this fact depends in an essential way
on the geometry of 8D since, as noted earlier, aD(z1, z2) = 0 whenever z 1 , z2 E D
are symmetric in a circle C which contains 8D.
Our argument makes use of the following sharp variant of the three-point con-
dition for a quasicircle. See Corollary 2.24 in Gehring-Hag [58].

COROLLARY 10.4.12. If z1,z2,z3, z4 is an ordered quadruple of points on a

K -quasi circle C and if

max(lz1 - zol, lz3 - zol) :'S: a :'S: b :'S: min(lz2 - zol, lz4 - zol)

for some point z 0 E R^2 , then

(10.4.13) bj a :<S: >..(K)^112 ,

where

THEOREM 10.4.14. If DC R^2 is a K-quasidisk, then

hD(zi, z2) :'S: >..(K) aD(z1, z2)

for each pair of points z1, z2 E D with

(10.4.15) aD(z1, z2) :<S: >..(K)-^1!^2.

PROOF. Set m = >..(K)^112 2: 1 and choose distinct z 1 , z 2 E D which satisfy
(10.4.15). By p erforming a preliminary Mobius transformation we may assume
that z 1 = 0, z 2 = oo and that

Then

and

(10.4.16)

r - 1 = inf lz l > 0,

z rf;D

r + 1 = sup lz l < oo.

zrf;D

R^2 \DC {z: r - 1 < lz l < r + 1}

(

2/ r <log r+ r _ 1)

1

= aD(O, oo) :<S: 1/ m