The set of infinite valence values of an analytic function

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)|