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.