The set of infinite valence values of an analytic function

Abstract

It is shown (Theorem A and its corollary) that if g is any nonconstant nonunivalent analytic function on a half-plane H and if D is either a half-plane or a smoothly bounded Jordan domain, then there is a function f on D for which f'(D) subset g'(H) such that for any neighborhood U of any point of f(boundary D) the set of values w in U which f assumes infinitely many times in D has Hausdorff dimension 1. From this it follows (Theorem C) that in the Becker univalence criteria for the disc and upper half-plane (|f"(z)/f'(z)|<=1/(1-|z|^2) and |f"(z)/f'(z)|<=1/(2Im{z}), respectively) if the 1 in the numerator is replaced by any larger number, then there are functions f satisfying the resulting bounds the set of whose infinitely assumed values has this same dimension 1 property.