Criteria for bounded valence of harmonic mappings

TL;DR

This paper extends classical criteria for bounded valence from analytic to harmonic functions, providing new conditions based on the Schwarzian derivative and derivative ratios.

Contribution

It generalizes existing bounded valence criteria from analytic functions to harmonic functions, broadening the scope of these important geometric function theory results.

Findings

Generalized bounded valence criteria to harmonic functions.

Established conditions involving Schwarzian derivative for harmonic mappings.

Extended classical results to a broader class of functions.

Abstract

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 $\limsup_{|z|\to 1} |S(f)(z)| (1-|z|^2)^2 < 2$, 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 $\limsup_{|z|\to 1} |f"(z)/f'(z)|\, (1-|z|^2)< 1$. In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when $f$ is merely harmonic.