  Generalizing several previous results in the literature on rational harmonic
functions, we derive bounds on the maximum number of zeros of functions $f(z) =
\frac{p(z)}{q(z)} - \overline{z}$, which depend on both $\mathrm{deg}(p)$ and
$\mathrm{deg}(q)$. Furthermore, we prove that any function that attains one of
these upper bounds is regular.
