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Characterizing Meromorphic Pseudo-lemniscates

  • Trevor Richards
Article
  • 38 Downloads

Abstract

Let f be a meromorphic function with simply connected domain \(G\subset \mathbb {C}\), and let \(\Gamma \subset \mathbb {C}\) be a smooth Jordan curve. We call a component of \(f^{-1}(\Gamma )\) in G a \(\Gamma \)-pseudo-lemniscate of f. In this note, we give criteria for a smooth Jordan curve \(\mathcal {S}\) in G (with bounded face D) to be a \(\Gamma \)-pseudo-lemniscate of f in terms of the number of preimages (counted with multiplicity) which a given w has under f in D (denoted \(\mathcal {N}_f(w)\)), as w ranges over the Riemann sphere. As a corollary, we obtain the fact that if \(\mathcal {N}_f(w)\) takes three different value, then either \(\mathcal {S}\) contains a critical point of f, or \(f(\mathcal {S})\) is not a Jordan curve.

Keywords

Meromorphic functions Lemniscates Finite Blaschke products 

Mathematics Subject Classification

30D30 

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Copyright information

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

Authors and Affiliations

  1. 1.Mathematics and Computer Science DepartmentMonmouth CollegeMonmouthUSA

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