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Computational Methods and Function Theory

, Volume 18, Issue 3, pp 463–472 | Cite as

The Maximum Number of Zeros of \(r(z) - \overline{z}\) Revisited

  • Jörg LiesenEmail author
  • Jan Zur
Article

Abstract

Generalizing several previous results in the literature on rational harmonic functions, we derive bounds on the maximum number of zeros of functions \(f(z) = \frac{p(z)}{q(z)} - \overline{z}\), which depend on both \(\mathrm{deg}(p)\) and \(\mathrm{deg}(q)\). Furthermore, we prove that any function that attains one of these upper bounds is regular.

Keywords

Zeros of rational harmonic functions Rational harmonic functions Harmonic polynomials Complex valued harmonic functions 

Mathematics Subject Classification

30D05 31A05 37F10 

Notes

Acknowledgements

We thank an anonymous referee for several helpful suggestions, and in particular for pointing out the technical report [7].

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.TU Berlin, Institute of MathematicsBerlinGermany

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