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