On the coefficients of concave univalent functions

✍ Scribed by Farit G. Avkhadiev; Christian Pommerenke; Karl-Joachim Wirths

• Publisher: John Wiley and Sons
• Year: 2004
• Tongue: English
• Weight: 103 KB
• Volume: 271
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310177

✦ Synopsis

Abstract

Let 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 |a~n~| ≥ 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 a~2~ and a~3~ for these classes of functions. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)