Criteria for bounded valence of harmonic mappings

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 ${\mathrm{lim\hspace{0.17em}sup}}_{\mathrm{\mid }z\mathrm{\mid }\to 1}\mathrm{\mid }S(f)(z)\mathrm{\mid }(1-\mathrm{\mid }z{\mathrm{\mid }}^2{)}^2<2$, 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.