60 4. INJECTIVITY CRITERIA
The following shows that the size of the Schwarzian relative to the hyperbolic
metric is related to the global injectivity of a meromorphic function.
THEOREM 4.1.3 (Lehto [114], Nehari [137]). If f is meromorphic and injective
in a simply connected domain D , then
sup IS1(z)I Po(z)-^2 :::; 3.
zED
The constant 3 is sharp.
Thus the Schwarzian derivative Sf is bounded in absolute value by a multiple of
the square of the hyperbolic density p D whenever f is injective in a simply connected
domain D. It is natural to ask if there is a sufficient condition, corresponding to
this necessary condition, for injectivity.
This is indeed the case whenever D is a disk or half-plane.
THEOREM 4.1.4 (Nehari [137]). If D is a disk or half-plane and if f is mero-
morphic with
sup IS1(z)I Po(z)-^2 :::; 1/2,
zED
then f is injective in D. The constant 1 /2 is sharp.
It is hence natural to ask for which domains the above result holds with 1/2
replaced by some positive constant.
DEFINITION 4.1.5. We let u(D) denote the supremum of the constants a ~ 0
such that f is injective whenever f is meromorphic and locally injective in D with
(4.1.6) sup IS1(z)I Po(z)-^2 :::; a.
zED
Then u(D) :::; 1/2 for all domains D and D is a disk or half-plane if and only
if u(D) = 1 /2 (Lehtinen [112]).
We observe next that supremum may be replaced by maximum in the definition
of u(D).
THEOREM 4.1.7 (Lehto [114]). If f is meromorphic in a simply connected
domain D with
( 4.1.8) sup IS1(z)I PD(z)-^2 :::; u(D),
zED
then f is injective in D.
SKETCH OF PROOF. Suppose that f is meromorphic in D with
sup IS1(z)I po(z)-^2 :::; u(D),
zEDfix points z1, z2, z3 ED so that f(z1), f(z2), f(z3) are distinct, and choose rj E (0, 1)
so that r r~ l. Then for each j there exists a function fj meromorphic in D such
that
Sfj (z) = rj S1(z)
in D; see, for example , Lehto [116]. Moreover by (4 .1.2) we may assume that
fj(zk) = f(zk) fork= 1, 2, 3. Then
sup ISfj (z)I Po(z)-^2 :::; rj u(D) < u(D),
zED
the functions fj are injective in D , and a subsequence converges locally uniformly
to f. Hence f is also injective in D. D