8 CONWAY, HERRERO, AND MORREL
used in this paper, call an analytic function f completely non-constant if f is
not constant on any component of its domain.
1.2 Proposition Assume that f is analytic in a neighborhood of a(A) and
completely non-constant ; put T = f(A). If Q is an analytic Cauchy region
with c l Q c P±(T) and dQ n f(Z(f*)) = 0 , then f~l{Q) n P±(A) consists of a
finite non-zero number of components H1 , . . . , H
d
and for each j , f(Hj) = Q ,
f(3Hj) = 3Q , Hj is an analytic Cauchy region, and there is a natural number pj
such that f is a strictly pj-valent map of Hj onto Q .
Proof. Since Q is connected, there is an extended integer N such that ind
(X - T) = N for X in cl Q . Let H =
f'l[Cl)
n a (A) ; by (1.1) , H c a (A) \ alre(A)
and H n P+(A) * 0 . Note that H is bounded.
Claim 1. If D is a component of H , then f(D) = Q. .
If co G Q. \ f(D) , then there is a path in Q. from co to a point £ in f(D) . Look at
the first point £0 on this curve that is not in f(D). So £0 e Q n df(D) . Let
{ z
n
} c D such that f(zn) - !^0 . Since { z
n
} c a(A), we may assume that z
n
-
z
0
in a(A) . Hence z
0
e f_1(Q) n a(A) = H . But D u ( z
0
) is connected and
included in H . Since D is a component of H , z
0
e D . This implies £0 =
f(z0) G f(D) n 3f(D), a contradiction.
Claim 2. H has only a finite number of components.
Indeed, if not, then there is an co in Q. such that f(z) = co has an infinite
number of solutions in the compact set a(A).
Claim 3 . If D is a component of H , then f(3D) = 3Q .
By Claim 1, f(D) = Q . It is clear that f(3D) c c l Q . If co e 3Q = 3f(D) c cl f(D)
c f(cl D) , then co = f(z) for some z in cl D . If z e D, co G Q , a contradiction.
Hence dQ c f(3D).
Now let COQ = f(z0) for some z
0
in dD . If co0 G Q , then there is an e 0
Previous Page Next Page