1549055259-Ubiquitous_Quasidisk__The__Gehring_

110 8. FIRST SERIES OF IMPLICATIONS

for z E H U H * where

K = 2k(k + 1).

See, for example, Beurling-Ahlfors [23], Lehto-Virtanen [117]. Hence f is K-

quasiconformal by Theorem 1.3.12. A further calculation then yields (8.6.6) for

z1, z2 EH where

L = 4k^2 (k + 1).

See Ahlfors [6] for the details. 0

We now use Theorem 8.6.3 to show that a Jordan domain D is a quasidisk if

for all pairs of conjugate quadrilaterals Q and Q* with boundary in 8D, mod( Q*)

is bounded whenever mod( Q) = l.

THEOREM 8.6.7. Suppose Dis a Jordan domain in R

2

and suppose there exists

a constant c ::;:: 1 such that

(8.6.8) mod(Q*):::; c

whenever Q and Q* are conjugate quadrilaterals with boundary in 8D and

mod(Q) = l.

Then there exists a K-quasiconformal mapping h : R

2

-+R

2

such that D = h(H)

and such that

(8.6.9) f*(z) =ho r o h-^1 (z) where r(z) = z

defines a hyperbolic L-bilipschitz reflection in 8D where K = K(c) and L = L(c).

PROOF. Choose conformal mappings g : H-+D and g* : H*-+D*. Then g

and g* have homeomorphic extensions to H and H* and, by means of an auxiliary

Mo bi us transformation, we may arrange that g( oo) = g* ( oo). Hence

<f>(z ) = g*-^1 o g(z )

is a self-homeomorphism of R^1 with ¢(00) = oo.

Fix x E R and t > 0 and let

Z1 = X - t, Z2 = X, Z3 = X + t,

W1=<f>(x-t), W2 = <f>(x), W3=<f>(x+t),

Then Q 1 = H(z 1 , z 2 , z 3 , z 4 ) is a quadrilateral in H,

Z3 - Z1

[z 1 , z 2 , z 3 , z 4 ] = ---= 2,

Z3 - Z2

and hence

(8.6.10)

Z4 = OO,

by (3.10.5) and (3.10.6). Similarly Qi = H(w 4 , w 3 , w 2 , w 1 ) is a quadrilateral in
H )

where