2015/08/21 by Julian Gevirtz, Gevirtz, Julian
Mathematics · #Algebraic and Geometric Analysis #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics #advanced mathematical theories #math.CV
paper · pdf · doi:10.48550/arxiv.1508.05416
arxiv created 2015/08/21 · openalex publication_date 2015/08/21 · arxiv updated 2015/08/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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/(2Imz), 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.