Notes on the Level Curves of a Meromorphic Function
Abstract
The subject of this paper is the bounded level curves of a meromorphic function f with domain G such that each component of ∂G consists of a level curve of f. (A primary example of such a function being a ratio of finite Blaschke products of different degrees, with domain D.) We will first prove several facts about a single bounded level curve of a f in isolation from the other level curves of f. We will then study how the level curves of f lie with respect to each other. It is natural to expect that the sets \z:|f(z)|=ε\ vary continuously as ε varies. We will make this notion explicit, and use this continuity to prove several results about the bounded level curves of f. It is well known that if z0 is a zero or a pole of f, then f is conformally equivalent to the function zzk (for some k∈Z) in a neighborhood of z0. We generalize this fact by finding a natural decomposition of G into finitely many sub-regions (also bounded by level curves of f), on each of which f is conformally equivalent to zzk (for some k∈Z). Also included is a new proof, using level curves, of the Gauss--Lucas theorem that the critical points of a polynomial are contained in the convex hull of the polynomial's zeros.