62 4. IN JECTIVITY CRITERIA
THEOREM 4.1.11 (Osgood [141]). If f is analytic and injective in a simply
connected domain D C R^2 , then
sup ITJ(z) I PD(z)-^1 ::::; 4.
zED
The constant 4 is sharp.
For which domains D can we reverse the implication in Theorem 4.1.11?
DEFINITION 4.1.12. We let T(D) denote the supremum of the constants b 2 0
such that f is injective whenever f is analytic and locally injective in D C R^2 with
(4.1.13) sup ITJ(z) I PD(z) -^1 :Sb.
zED
Then T(D) ::::; 1/2 for all domains D (Stowe [155]) and equality holds whenever
D is a disk or half-plane (Becker-Pommerenke [22]). However in this case the
converse does not hold; there exists a domain D with T(D) = 1/2 which is not a
disk or half-plane (Stowe [155]).
On the other hand, we have the following analogue for Theorem 4.1.9.
THEOREM 4.1.14 (Astala-Gehring [14]). A simply connected domain D is a
K-quasidisk if and only if T(D) > 0, where K and T(D) depend only on each other.
Though the constants a(D) and T(D) reflect the geometry of the domain D ,
little is known about their values except for a(D) in the following special cases.
1° If D is a sector of angle a, then
1 ( a^2 2 7r a - a
2
a(D) = - min - )
2 7r^2 , 7r^2.
2° If D is a regular n-sided polygon, then3° Finally, if D is a rectangle with side ratio r, then
a(D) = 1/8
for .65 7 < r < 1.523. In particular, a(D) does not always depend analyt-
ically on the shape of D.See Lehto [115] for 1°, see Calvis [30], Lehtinen [113] for 2°, and see Miller-Van
Wieren [131] for 3°.
4.2. Locally bilipschitz mappings
We consider next some analogues for bilipschitz functions of the injectivity and
extension results established in the previous section.
DEFINITION 4.2.l. The mapping f is a locally L-bilipschitz mapping of E c R^2
if each point of E has a neighborhood U such that
1
L lz1 - z2I :S lf(z1) - f(z2)I :SL lz1 - z2Ifor Z1) Z2 E E n u.
Suppose that f is a locally L-bilipschitz mapping of D. Then whether or not
one can conclude that f is injective depends on L and on D.