  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.
