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

Abstract

An important theorem of Khavinson and Neumann (Proc. Am. Math. Soc. 134: 1077–1085, 2006) states that the complex harmonic function $$r(z) - \overline{z}$$ , where $$r$$ is a rational function of degree $$n \ge 2$$ , 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 $$r(z) - \overline{z}$$ no more than $$5 (n - 1) - 1$$ zeros can occur. Moreover, we show that $$r(z) - \overline{z}$$ is regular, if it has the maximal number of zeros.