1549055259-Ubiquitous_Quasidisk__The__Gehring_

8.6. QUADRILATERAL INEQUALITY AND QUASIDISKS 109

PROOF. Set f(z) = f(z) for z E H*. Then f is a self-homeomorphism of R^2

which is K-quasiconformal in H , in H*, and hence in R^2 by Theorem 1.3.12. Next

l(x+t)-xl = lx-(x-t)I and

lf(x + t) - f(x)I::; k lf(x) - f(x - t)I,

lf(x) - f(x - t)I::; k lf(x + t) - f(x)J

by Theorem 1.3.4 from which we obtain (8.6.2) with k = e^8 K.

The sharp estimate for the constant k in (8.6.2) is k = >..(K) where

>..(K) = ( ~ e"K/^2 - e-"K/^2 )

2

+ 6(K), 0 < 6(K) < e-"K,

as in (1.3.6).

D

Functions ¢ : I C R^1 ~R^1 which satisfy (8.6.2) for relevant x and t are said
to be k-quasisymmetric (Kelingos [100]). They are, in a sense, one-dimensional
quasiconformal mappings. On the real line this notion of quasisymmetry is equiv-
alent to a condition which in general is strictly stronger and which can be used
to define quasisymmetric mappings between metric spaces. See Heinonen [80] and
also Astala-Iwaniec-Martin [16].
Lemma 8.6.1 implies that the boundary mapping induced by a quasiconformal
self-mapping of His quasisymmetric. The following important result due to Beurl-
ing and Ahlfors [23] shows that t he converse is true. See also Ahlfors [6], Lehto
[116], and Lehto-Virtanen [117].

THEOREM 8.6.3. Suppose that¢: R^1 ~R^1 is a homeomorphism and that

(8.6.4)
1 ¢(x+t)-¢(x) k

• < <
  k - ¢(x) - ¢(x - t) -

for all real x and t > 0. Then there exists a K-quasiconformal mapping f: R
2
~R
2

with

(8.6.5) f(x) = ¢(x)

for x E R^1 where K = K ( k). Moreover f is L-bilipschitz with respect to the
hyperbolic metric in H , i.e.,

(8.6.6)

for z1, z2 EH where L = L(k).

SKETCH OF PROOF. We may assume without loss of generality that ¢(t) is
increasing int. For z = x + iy E R^2 set

1 i

f(z ) = u(z) + iv(z) =

2

(a(z) + f3(z)) +

2

(a(z) - f3(z))

where

and f3(z) = fo

1

¢(x - ty) dt.

Then a technical argument based on inequality (8.6.4) shows that f is a homeo-
morphism and that

H (z ) = 1 + lμ1(z)I < K

(^1) 1 - lμ1(z) I -