Commentarii Mathematici Helvetici

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Volume 81, Issue 4, 2006, pp. 801–807
DOI: 10.4171/CMH/74

Published online: 2006-12-31

Sharp inequalities for the coefficients of concave schlicht functions

Farit G. Avkhadiev[1], Christian Pommerenke[2] and Karl-Joachim Wirths[3]

(1) Kazan State University, Russian Federation
(2) Technische Universit├Ąt Berlin, Germany
(3) Universit├Ąt Braunschweig, Germany

Let $D$ denote the open unit disc and let $f\colon D\to \mathbb{C}$ be holomorphic and injective in $D$. We further assume that $f(D)$ is unbounded and $\mathbb{C}\setminus f(D)$ is a convex domain. In this article, we consider the Taylor coefficients $a_n(f)$ of the normalized expansion $$f(z)=z+\sum_{n=2}^{\infty}a_n(f)z^n, z\in D,$$ and we impose on such functions $f$ the second normalization $f(1)=\infty$. We call these functions concave schlicht functions, as the image of $D$ is a concave domain. We prove that the sharp inequalities $$|a_n(f)-\frac{n+1}{2}|\leq\frac{n-1}{2}, n\geq 2,$$ are valid. This settles a conjecture formulated in [2].

Keywords: Taylor coefficients, concave schlicht functions, slit mappings

Avkhadiev Farit, Pommerenke Christian, Wirths Karl-Joachim: Sharp inequalities for the coefficients of concave schlicht functions. Comment. Math. Helv. 81 (2006), 801-807. doi: 10.4171/CMH/74