Characterizing Meromorphic Pseudo-lemniscates

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.

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Correspondence to Trevor Richards.

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Communicated by Dmitry Khavinson.

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Richards, T. Characterizing Meromorphic Pseudo-lemniscates. Comput. Methods Funct. Theory 18, 609–616 (2018). https://doi.org/10.1007/s40315-018-0242-6

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Keywords

  • Meromorphic functions
  • Lemniscates
  • Finite Blaschke products

Mathematics Subject Classification

  • 30D30