1549055259-Ubiquitous_Quasidisk__The__Gehring_

12 1. PRELIMINARIES

1.4. Quasidisks

We come now to the principal object of study in this book.

DEFINITION 1.4.1. A domain Dis a K-quasidisk if it is the image of a Euclidean

disk or half-plane under a K-quasiconformal self-mapping of R

2

. D is a quasidisk
if it is a K-quasidisk for some K.

We present next three Jordan domains that we will use in what follows to
illustrate various properties of quasidisks. The first of these is an angular sector.

EXAMPLE 1.4.2. For 0 < a < 27f let S(a) denote the angular sector

S(a) = { z = rei^6 : 0 < r < oo, IBI < ~}.

Then S(a) is a K-quasidisk where

(1.4.3)

(

K=max y-;-'V~ ~~).

The bound in (1.4.3) is sharp.

To prove this, let

f(rei 6) =rPei<f>(6)

for 0 < r < oo and IBI :::; 7f where

7f

p= ---;====

J(27r - a) a

and

¢(0) ~!

7f e

if o:::; e:::; ~,

a

7r(7r-B)

if

a

7f - - < e < 7f,

27f - a 2 - -

-¢(-B) if -7f:::; e:::; o.

An elementary calculation shows that f is K-quasiconformal, where K is as in

(1.4.3), and that f maps S(a) onto the right half-plane S(7r).

To show that the bound in (1.4.3) is best possible, suppose that f is a K -

quasiconformal mapping ofR

2
which maps S(a) onto the right half-plane S(7r) and
let h = 1-^1 o g o f where g denotes reflection in the imaginary axis. Then h is a

K^2 -quasiconformal mapping of R

2
which maps S(a) onto its exterior S*(a).
Next fix 0 < a < b < oo and let r denote the family of arcs which join the
circles {z: lz l =a} and {z: lz l = b} in

{z : a :S lz l :Sb, I arg(z ) I < a/2}.

Then it is not difficult to check that
a
mod(r) = log(b/a) ·
Similarly,
mod(r') =^2 7f - a
log(b/a) + 2 log(e)
where r' is the family of arcs which join { z : lz l = a/ e} and { z : lz l = be} in

{z: a/ e :S lz l :S be, a / 2 < I arg(z)I :S 7f }.