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

Abstract

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 $n\ge 2$, 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 $r(z)-\bar{z}$ 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.

Some minor corrections and extensions

References in corpus (2)

• Perturbing rational harmonic functions by poles
• Sharp parameter bounds for certain maximal point lenses

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