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.
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Acknowledgments
We would like to thank Dmitry Khavinson for pointing out reference [1] to us. We are grateful to the anonymous referee for the careful reading of the manuscript, and many valuable remarks.
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Communicated by Dmitry Khavinson.
The work of R. Luce was supported by Deutsche Forschungsgemeinschaft, Cluster of Excellence “UniCat”.
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Luce, R., Sète, O. & Liesen, J. A Note on the Maximum Number of Zeros of \(r(z) - \overline{z}\) . Comput. Methods Funct. Theory 15, 439–448 (2015). https://doi.org/10.1007/s40315-015-0110-6
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DOI: https://doi.org/10.1007/s40315-015-0110-6