10 CONWAY, HERRERO, AND MORREL
Claim 7 If D is a component of H , then D is an analytic Cauchy region.
In fact, this is clear since Q is an analytic Cauchy region and f does not
vanish on 3H .
This completes the proof of the proposition.
Some remarks and examples relative to the preceding proposition
might be helpful for the reader. Let f and Q. be as in Proposition 1.2 and let
D be one of the components of f_1(^) ^ P+(A) such that f is a strictly p-valent
mapping of D onto Q. . Suppose 3Q consists of m pairwise disjoint analytic
Jordan curves yl , . . . , y
m
and 3D consists of n pairwise disjoint analytic
Jordan curves gx , . . . , g
n
. Then f maps the boundary curve gj onto some
component curve of 3Q in an Tj -to-one fashion and, moreover, for 1 i m
X {ij : fig,) = Yi) = P :
consequently,
n
X
r
J
= mp

j = l
The following examples will convince the reader that, except for these
equalities, everything else is possible.
If f(z) = z p and Q = D = D , then f maps 3D p-to-one onto dQ. . At the
other extreme, if D is an analytic Cauchy region whose boundary consists of
p pairwise disjoint analytic curves gx , . . . , gp , let f be the Ahlfors function
mapping D onto Q. = D . (See [1].) Then f is a strictly p-valent function on Q
and f is a bijection of each gj onto 3D .
For a further example, let f be the monic polynomial with distinct
zeros ^ , . . . , Xs and let X{ have multiplicity di ; put p = d
1
+ - - - + d
s
=
degree of f . Let
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