CHAPTER 4Injectivity criteria
Suppose that 1 is a mapping which is locally injective in a domain D. When
can we conclude that 1 is globally injective in D? The answer depends, of course,
on the nature of the function 1 as well as on the domain D.
We consider here three classes of functions 1 as well as appropriate measures of
growth which are necessary and which are sufficient for the functions in each class
to be injective whenever D is a disk or half-plane. We then observe that the same
kinds of conclusions hold if and only if D is a quasidisk.
We conclude with a characterization of the Jacobian of a conformal mapping
1 of D which holds whenever D is a quasidisk.
4.1. Meromorphic functions
We adopt the convention that a meromorphic function need not have poles
and may be defined in a neighborhood of oo and that an analytic function is a
meromorphic function without poles defined in a domain in R^2. We consider here
two criteria for the injectivity of a function f in these two classes involving
1 ° the Schwarzian derivative Sf of f,
2° the pre-Schwarzian derivative Tt of f.We then show how each of these criteria can be used to characterize quasidisks.
The Schwarzian derivative of a function 1 meromorphic and locally injective in
Dis given by
= (1")' -~ (1")2
St l' 2 !' '
where we employ the usual convention regarding points in { oo, 1-^1 (oo )}.
If f is a Mobius transformationf ( z ) = a z + b , ad - be =/: 0,
cz+d
then
f"(z) 2c (f"(z))' 2c^2
l'(z) - c z + d' l'(z) (cz + d)^2
and hence S1(z) = 0 for all z. Conversely if 1 is meromorphic with S1(z) = 0 in a
domain D, then f is the restriction of a Mobius transformation to D in which case
1 is injective in D.
If 1 is meromorphic in D and g is meromorphic in f(D), then
(4 .1.1) Sgof(z) = S 9 (f(z)) f'(z )^2 + S1(z)
in D. In particular, whenever g is a Mobius transformation,
(4 .1.2)
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