1549055259-Ubiquitous_Quasidisk__The__Gehring_

38 3. CONFORMAL INVARIANTS

where w 1 = r e-i et./^2 and w 2 = r ei et./^2 , and with (3.2.2) we obtain

1 1

112

an(z ( a. -^2 e a. +^2 e)

1 , z 2 ) ~ - cot ---+cot -- 4 - de

2 111 4

= rll 2 sin ( a./2) de

1 11, cose - cos (a./2)

Suppose next that Zj = rj eill where 0 < r1 < r2 and - a./2 < e < a/2. If
8 = 0, then (3.3.11) with W1 = 0 and W2 = 00 yields

A more detailed calculation yields the same result when B =f. 0. These estimates
are asymptotically sharp in each case as hn(z1, z2)---+oo.

From Example 3.3.9 it follows that

2n 2n

hn(z1,z2):::; -an(z1,z2):::; -fo(z1,z2)

a. a.

for z 1 , z 2 E D when D is a sector of angle a. with 0 < a :::; n. The following results
show that similar bounds hold whenever D is a quasidisk.

THEOREM 3.3. 12 (Jones [94]). A simply connected domain D is a K-quasidisk

if and only if there exists a constant c such that

hn(z1, z2):::; cfo(z1, z2)

for z 1 , z2 E D , where K and c depend only on each other.

We see from Theorem 3.3.12 and Lemma 3.3.5 that D is a quasidisk if and
only if the conformally invariant metric hn is equivalent to the similarity invariant
metric Jn in D.
This is also the case if and only if hn is equivalent to the Mobius invariant
metric an in D.

THEOREM 3.3. 13 (Gehring-Hag [58]). A simply connected domain Dis a K-

quasidisk if and only if there exists a constant c such that

(3.3.14)

for z 1 , z 2 E D, where K and c depend only on each other.

We conjecture that a bounded simply connected domain D is a disk if it satisfies
(3.3.14) with c = l. This is the case if, in addition, for each z E 8D there exists a
disk D' = D'(z ) CD with z E 8D'. See Gehring-Hag [56].