1549055259-Ubiquitous_Quasidisk__The__Gehring_

3.10. QUADRILATERALS 49

To show that these bounds are sharp, suppose that 0 < a :::; 7r and for 1 <

r < oo let Q = D(z1, z2, z3, z4) denote the quadrilateral in D = S(a) with vertices

z1 = reia./^2 , z2 = eia./^2 , z3 = e-ia./^2 , z 4 = re-ia./^2 and set

7r

p= -,

a

7r

q=.

27r-a

Then f(z) = i zP maps Q conformally onto Q' = H(w 1 , w 2 , w 3 , w 4 ) where w 1 = -rP,

W2 = -1, W3 = 1, w4 = rP. Hence by Lemma 3.10.4

mod(Q) = mod(Q') = -^2 μ (2r-P/ '.::::'. - log r

2

) p

7r 1 + r-P 7r

as r-+oo. Similarly

mod(Q*) = -μ^2 ( 2r-q/^2 ) '.::::'.-log q r

7r 1 + r-q 7r

as r-+oo,

lim mod(Q*) = 27r - a

r--+oo mod(Q) a '

and hence the upper bound in Example 3.10.7 is sharp. A similar argument yields

the same result for the lower bound.

If Dis a sector of angle a, then Example 3.10.7 implies that

mod( Q*) :::; max (^2 7r - a, _a __ )

a 27r - a

whenever Q* is a quadrilateral conjugate to Q with respect to 8D with mod(Q) = 1.

We are thus led to one of the first published criteria for quasidisks (Tienari [159],

Lehto-Virtanen [117], Pfluger [144]).

DEFINITION 3.10.8. A Jordan domain D satisfies the conjugate quadrilateral

inequality if there exists a constant c ::::: 1 such that if Q and Q* are conjugate

quadrilaterals with respect to 8D and if mod(Q) = 1, then

mod(Q*):::; c.

The following example shows that the above property characterizes disks and

half-planes when c = 1.

EXAMPLE 3.10.9. A Jordan domain D is a disk or half-plane if and only if it
satisfies the conjugate quadrilateral inequality with c = 1.

To establish the sufficiency, suppose that f and g map D and D conformally
onto the upper and lower half-planes H and H, respectively. Then f and g have
homeomorphic extensions to D and D* and ¢ = f o g-^1 is a self-homeomorphism

of R

1

. By renormalizing we may assume that ¢ fixes the points 0, 1, oo.
The quadrilaterals Q = H(a, b, c, oo) and Q = H(¢(a), ¢(b), ¢(c), oo) have
modulus 1 if and only if

b-a=c-b and ¢(b) - ¢(a)= ¢(c) - ¢(b),

respectively. By hypothesis mod(Q*) = 1 whenever mod(Q) = 1. It then follows
by induction that ¢(x) = x, first for x = m where m = 0, ±1, ±2,... and then for