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An important theorem of Khavinson & Neumann (Proc. Amer. Math. Soc. 134(4),2006) states that the complex harmonic function $r(z) - \bar{z}$, where $r$ isa rational function of degree $n \geq 2$, has at most $5 (n - 1)$ zeros. Inthis note we resolve a slight inaccuracy in their proof and in addition we showthat 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 ithas the maximal number of zeros.

A Note on the Maximum Number of Zeros of $r(z) - \bar{z}$