On the coefficients of concave univalent functions

Abstract

AbstractLet D denote the open unit disc and f : D → \documentclass{article} \usepackage{amssymb} \pagestyle{empty} \begin{document} $ \overline {\mathbb C} $ \end{document} be meromorphic and injective in D. We assume that f is holomorphic at zero and has the expansion Especially, we consider f that map D onto a domain whose complement with respect to \documentclass{article} \usepackage{amssymb} \pagestyle{empty} \begin{document} $ \overline {\mathbb C} $ \end{document} is convex. We call these functions concave univalent functions and denote the set of these functions by Co.We prove that the sharp inequalities |an| ≥ 1, n ∈ ℕ, are valid for all concave univalent functions. Furthermore, we consider those concave univalent functions which have their pole at a point p ∈ (0, 1) and determine the precise domain of variability for the coefficients a2 and a3 for these classes of functions. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)