1549055259-Ubiquitous_Quasidisk__The__Gehring_

88 7. MISCELLANEOUS PROPERTIES

and f(z) = z, then g(z) = ez is conformal in D , f(z) =log g'(z), and

llfllB(D):::; 8
by what was proved above. However when x > 1,
lf(x) - f(l)I Ix - ll
-----= -+oo
]v(x, 1) 2 log lx l
as x -+oo.

The simply connected domains D where Bloch functions f have a multiple of

the metric JD as a modulus of continuity are quasidisks.

THEOREM 7.1.4 (Langmeyer [110]). A simply connected domain DC R^2 is a
quasidisk if and only if there exists a constant a > 0 such that
lf(zi) - f(z2)I :::; a ll!llB(D) ]v(z1, z2)
for each f in B(D) and z 1 , z2 ED.
The above result suggests t he following question. For what kinds of conforma l
mappings g(z) are the functions in (7.1.2) Bloch functions? The answer yields
another characterization for quasidisks.

THEOREM 7.1.5 (Broch [28]). A simply connected domain D c R^2 is a qua-

sidisk if and only if there exists a constant b > 0 such that each f in B(D) with

llfllB(D) :::; b is of the form

f ( z) = log g' ( z)

where g is conj ormal in D.

PROOF. Suppose t hat f is analytic in D and let

g(z) = r ef(()d(

}zo

where z 0 E D. Then g is analytic with

f(z) = logg'(z),

I g" (z)

f (z) = g'(z) = T 9 (z)

for z ED and
1

• 2

sup IT 9 (z)IPD(z)-^1 :::; llJllB(D):::; 2sup IT 9 (z)IPD(z)-^1

D D

by (3.2.1) and (7.1.1).
If Dis a quasidisk, then by Theorem 4.1. 14 there exists a constant c > 0 such
that g is conformal in D whenever

sup IT 9 (z)IPD(z)-^1 :::; 2c

D

and hence whenever llJllB(D) :::; c.
Conversely if there exists a constant c > 0 such that f = logg^1 is in B(D) with
g conformal in D whenever llJllB(D) :::; c and hence whenever

c

s~p IT 9 (z)lpD(z)-^1 :::; 2'

then Dis a quasidisk by Theorem 4.1.14. D