2016/11/17 by Juha-Matti Huusko, María J. Martín, Huusko, Juha-Matti +1
Mathematics · #30C55 #31A05 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.1611.05667
openalex publication_date 2016/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S(f) of a locally univalent analytic function f in the unit disk satisfies that \limsup|z|→ 1 |S(f)(z)| (1-|z|2)2 < 2, then there exists a positive integer N such that f takes every value at most N times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function f satisfies that \limsup|z|→ 1 |f"(z)/f'(z)| (1-|z|2)< 1. In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when f is merely harmonic.