The set of infinite valence values of an analytic function

TL;DR

This paper investigates the set of values an analytic function assumes infinitely often near boundary points, showing that under certain conditions, this set has Hausdorff dimension 1, revealing new insights into univalence criteria.

Contribution

It establishes that for nonunivalent analytic functions, the set of infinitely assumed boundary values has Hausdorff dimension 1, extending univalence criteria with new bounds.

Findings

The set of infinitely assumed boundary values has Hausdorff dimension 1.

Functions satisfying stronger Becker univalence bounds also have boundary value sets of Hausdorff dimension 1.

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.