  An important theorem of Khavinson & Neumann (Proc. Amer. Math. Soc. 134(4),
2006) states that the complex harmonic function $r(z) - \bar{z}$, where $r$ is
a rational function of degree $n \geq 2$, has at most $5 (n - 1)$ zeros. In
this note we resolve a slight inaccuracy in their proof and in addition we show
that for certain functions of the form $r(z) - \bar{z}$ no more than $5 (n - 1)
- 1$ zeros can occur. Moreover, we show that $r(z) - \bar{z}$ is regular, if it
has the maximal number of zeros.
