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

TL;DR

This paper refines the proof of a theorem on the maximum zeros of harmonic functions of the form r(z) - ar{z} and establishes conditions for achieving this maximum.

Contribution

It corrects an inaccuracy in the existing proof and shows that for certain functions, the maximum number of zeros is actually one less, providing a more precise understanding.

Findings

Corrected the proof of the maximum zeros theorem.

Established that some functions have at most 5(n - 1) - 1 zeros.

Proved that functions with the maximum number of zeros are regular.

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 $n \geq 2$, 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.