1549055259-Ubiquitous_Quasidisk__The__Gehring_

11.1. HARMONIC BENDING AND QUASIDISKS 151

D /z2

___l__

...-; -...... ...... /

Z1 ---_.. / -_.. ------~·-_.,_ ---··-----

lg

.1 in n+in

n

0 f(z ) 1

lh

/1

_.r. f(l)

_L - - - - --.. - - - -- --H - - -

Hence

(11.1.5)

-^1 -in.
n n-in

(ho f)(z)

FIGURE 11.2

2 ( 1 ) 4 ( 1 ) l/2

= ;: arctan lf(zo)I = ;: arctan ~

and (11.1.2) follows from (11.1.3) and (11.1.4). D

The above argument shows that there exists an example for which inequality in

Lemma 11.1.1 is sharp whenever diam(!) :::; dist(zo, 1) and that the simpler bound

(11.1.6) w(zo,1;D):::;-^4 ( d't( diam(!) ) ) ~

7r lS zo, I

holds for each arc/ in aD and each point zo ED.

We turn now to the proof of the following characterization for quasidisks in

terms of harmonic bending.

THEOREM 3.9.3 (Fernandez-Hamilton-Heinonen [41]). A Jordan domain Dis

a K -quasidisk if and only if it has the harmonic bending property with constant c,

where K and c depend only on each other.

PROOF. For necessity assume that C = aD is a quasicircle. Let/ be a closed
arc of C with endpoints z 1 , z2 and consider z E C \ /· Then there exists a K -
-2 -
quasiconformal map f of R which maps C onto Rand C \/onto (0, 1). In order
to use Definition 3.9.2 of harmonic bending we consider a preliminary conformal