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

Autor: Olivier Sète, Jörg Liesen, Robert Luce
Rok vydání: 2015
Předmět:
Zdroj: Computational Methods and Function Theory. 15:439-448
ISSN: 2195-3724
1617-9447
DOI: 10.1007/s40315-015-0110-6
Popis: 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.
Databáze: OpenAIRE
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