1549055259-Ubiquitous_Quasidisk__The__Gehring_

CHAPTER 7

Miscellaneous properties

We conclude our account of ways to describe a quasidisk D with some seem-

ingly unrelated characterizations. These include the continuity of Bloch functions,

homogeneity properties of D and f) D, Dirichlet integrals of functions harmonic in D

and D*, a description of the family of all quasicircles, and finally a relation between

quasiconformal mappings in R

2

and in R

3

.

7 .1. Bloch functions

A function f analytic in D c R^2 is said be a Bloch function, or is in B(D), if

(7.1.1) llfllB(D) =sup lf'(z)I dist(z, fJD) < oo.

zED

Bloch functions play an important role in complex analysis. For example, if

(7.1.2) f(z) =log g'(z)

where g is conformal in a simply connected domain D C R^2 , then

J'(z) = T 9 (z)

where T 9 is the pre-Schwarzian derivative of g. Hence f is in B(D) with

llJllB(D) =sup IT 9 (z)I dist(z, fJD)::; 2 sup IT 9 (z)I PD(z)-^1 ::; 8

zED zED

by inequality (3.2.1) and Theorem 4.1.11. For other examples see Bonk [26] and

Liu-Minda [119].

The bound for f'(z) in (7.1.1) implies that functions fin B(D) have the fol-

lowing modulus of continuity when D is a disk or half-plane.

EXAMPLE 7.1.3. If f is in B(D) where Dis a disk or half-plane, then

lf(zi) - f(z2)I::; llJllB(D) fo(z1, z2)

for z 1 , z 2 ED, where JD is the distance-ratio metric defined in (3.3.2).

Note that if D is a disk or half-plane, then

lf'(z) I::; llJllB(D) dist(:, fJD) ::; llfllB(D) PD(z)

for z ED and

lf(z1) - f(z2)I::; llJllB(D) hD(z1, z2)::; llJllB(D) fo(z1, z2)

by Example 3.3.7.

This result does not hold for all domains D. For example if

D = {z = x + iy: 0 < x < oo, IYI < 1}

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