1549055259-Ubiquitous_Quasidisk__The__Gehring_

50 3. CONFORMAL INVARIANTS

r

FIGURE 3.3

x = m 2-n where n = 0, 1, 2, .... Hence ¢(x) = x for all x E R^1 by continuity and

h(z) = {f(z)

g(z)

if z ED,

if z ED*

defines a Mobius transformation which maps D onto H.

THEOREM 3.10.10 (Tienari [159], Lehto-Virtanen [117], Pfluger [144]). A Jor-
dan domain D is a K -quasidisk if and only if it satisfies the conjugate quadrilateral
inequality with constant c, where K and c depend only on each other.

3.11. Extremal distance property
We observed earlier that mod(r) is a conformally invariant outer measure of a
family of curves which is large if the curves in r are short and plentiful and which
is small if the curves are long or scarce. We now use this quantity to compare
distances between two continua C 1 , C2 C D, as measured by the moduli of the

families of curves which join them in D and in R

2
, respectively. This, in turn,
leads to another characterization for quasidisks.
Given continua C1) C2 c D) we denote by r D and r the families of all curves
which join C 1 and C 2 in D and R
2
, respectively. We find it convenient to introduce
the notation

μD(C1, C2) = mod(I'D)

Since rD c r,

and

μD(C1, C2) :S: μ(C1, C2)

for all domains D. On the other hand, for certain domains D we can, up to a
constant, reverse the direction of the above inequality.

EXAMPLE 3.11.l (Yang [169]). If Dis a sector of angle a , then

27!'

(3.11.2) μ(C1, C2) :S: - μD(C1, C2)

a