On the Residuum of Concave Univalent Functions

Abstract

Let D denote the open unit disc and f:D→[`C] be meromorphic and injective in D. We further assume that f has a simple pole at the point p О (0,1) and is normalized by f(0) = 0 and f′(0) = 1. In particular, we are concerned with f that map D onto a domain whose complement with respect to [`C] is convex. Because of the shape of f(D) these functions will be called concave univalent functions with pole p and the family of these functions is denoted by Co(p). We determine for fixed p ∈ (0,1) the set of variability of the residuum of f, f ∈ Co(p).