1549055259-Ubiquitous_Quasidisk__The__Gehring_

8.6. QUADRILATERAL INEQUALITY AND QUASIDISKS 111

and thus

(8.6.11) mod(Qr)='!:_μ(

1

) ·

7r Jl + l /u

Let Q = g(Q1) and Q* = g*(Qr). Then Q and Q* are conjugate quadrilaterals

in D and D* and

mod(Q) =mod( Qi)= 1

by (8.6.10). Hence

mod(Q*) = mod(Qr):::; c

by (8.6.8) and

-μ^2 (^1 ) < c

n Jl + l /u -

by (8.6.11).
The above argument, with Q1 and Qi replaced by Q 2 = H(z 2 , z 3 , z 4 , zi) and
Q2 = H*(w1, W4, w3, w2), implies that

'!:_μ(l+u) :::;c.

7r 1 + u

Then since

lim μ(r) = oo,

r--+0

there exists a constant k = k(c) 2 1 such that

(8.6.12)
1 ¢(x+t)-¢(x) k

• < <
  k - ¢(x) - ¢(x - t) - ·
  Now (8.6.12) holds for all x ER and t > 0. Hence by Theorem 8.6.3 there exists
  -2 - 2
  a K-quasiconformal mapping f: R --+R where K = K(k) such that f(H) = H
  and f(x) = ¢(x) for x E 8H. In addition,

1
(8.6.13) L hH(w1, w2) :::; hH(f(w1), f(w2)) :::; L hH(w1, w2)

for w 1 , w 2 EH where L = L(k). Then

(8.6.14) h(z) = { go 1-l(z)
g*(z)

if z EH,

if z E H*

defines a self-homeomorphism of R
2
which is K-quasiconformal in Hand conformal

in H *. Thus his K-quasiconformal in R
2
and D = h(H) is a K-quasidisk.
Finally, set

(8.6.15) f(z) =ho r o h-^1 (z) = g or of o g-^1 (z)

for z ED where r(z) = z. Then J(D) = D, f*(z) = z for z E aD, and

hD·U*(z1), f*(z2)) hH(f(w1), f(w2))

hD(z1, z2) hH(w1, w2)

for z 1 , z 2 E D where Wj = g-^1 (zj) E H. We conclude from (8.6.13) that J* is a
hyperbolic L-bilipschitz reflection in aD. D