1549055259-Ubiquitous_Quasidisk__The__Gehring_

48 3. CONFORMAL INVARIANTS

a strictly decreasing homeomorphism of (0, 1) onto (0, oo). Furthermore, we have
that
4
μ(r) '="log -
r
as r --7 0 in (3.10.6).
SKETCH OF PROOF. Let f : H--+H be a Mobius transformation which maps
z 1 , z 2 , z 3 , z 4 onto -1/r, -1, 1, l/r, respectively, where 0 < r < l. Then

and the transformation

. g(w) =
1

w W,,;

o J(l -w^2 )(l -r^2 w^2 )

maps the quadrilateral Q' = H(-1/r, -1, 1, 1/r) conformally onto the quadrilateral

Q" = R(-K + iK' , -K, K, K + iK') so that vertices correspond, where

K = JC(r) and K'=JC(~).

Then

mod(Q) = mod(Q') = mod(Q") = K'^1 2 ( 2,jr)

2

K = ; μ(r) = ; μ

1

+ r

_'}:_μ( 1 )

• 7r J[z1,Z2,Z3,Z4].

See Anderson-Vamanamurthy-Vuorinen [11], [12] and Lehto-Virtanen [117]. D

Suppose next that C is a Jordan curve which bounds the domains D and D
and that z 1 , z 2 , z 3 , z 4 E C is a quadruple of points positively oriented with respect
to D. Then the quadrilaterals Q = D(z1,z2,z3,z4) and Q = D(z4,Z3,z2,z1) are
said to be conjugate with respect to C.
In particular if D is a disk or half-plane and if Q and Q are quadrilaterals
conjugate with respect to /JD, then
mod(Q*) = mod(Q).

The following example shows what to expect when D is a quasidisk.

EXAMPLE 3.10.7. If Dis a sector of angle a and if Q and Q* are quadrilaterals

conjugate with respect to /JD, then

< mod(Q*) <max (2w-a ,---a )

• mod(Q) - a 2w-a
  mm. (2w-a , a )
  a 2 w - a
  and these bounds are sharp.

To see this, suppose that D = S(a). By the discussion in Example 1.4.2 there

exists a K-quasiconformal self-mapping of R^2 which maps S(a) onto S*(a) where

K= max (2w-a , a ).

a 2w - a

Hence the above bounds follow from Theorem 1.2.2 and Lemma 3.10.l.