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Characterizing Meromorphic Pseudo-lemniscates

Characterizing Meromorphic Pseudo-lemniscates Let f be a meromorphic function with simply connected domain $$G\subset \mathbb {C}$$ G ⊂ C , and let $$\Gamma \subset \mathbb {C}$$ Γ ⊂ C be a smooth Jordan curve. We call a component of $$f^{-1}(\Gamma )$$ f - 1 ( Γ ) in G a $$\Gamma $$ Γ -pseudo-lemniscate of f. In this note, we give criteria for a smooth Jordan curve $$\mathcal {S}$$ S in G (with bounded face D) to be a $$\Gamma $$ Γ -pseudo-lemniscate of f in terms of the number of preimages (counted with multiplicity) which a given w has under f in D (denoted $$\mathcal {N}_f(w)$$ N f ( w ) ), as w ranges over the Riemann sphere. As a corollary, we obtain the fact that if $$\mathcal {N}_f(w)$$ N f ( w ) takes three different value, then either $$\mathcal {S}$$ S contains a critical point of f, or $$f(\mathcal {S})$$ f ( S ) is not a Jordan curve. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Characterizing Meromorphic Pseudo-lemniscates

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References (16)

Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-018-0242-6
Publisher site
See Article on Publisher Site

Abstract

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

Journal

Computational Methods and Function TheorySpringer Journals

Published: Apr 13, 2018

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