Criteria for bounded valence of harmonic mappings

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

10 pages