Characterizing Meromorphic Pseudo-lemniscates

摘要

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