# Sharp inequalities for the coefficients of concave schlicht functions

### Farit G. Avkhadiev

Kazan State University, Russian Federation### Christian Pommerenke

Technische Universität Berlin, Germany### Karl-Joachim Wirths

Universität Braunschweig, Germany

## Abstract

Let $D$ denote the open unit disc and let $f:D→C$ be holomorphic and injective in $D$. We further assume that $f(D)$ is unbounded and $C∖f(D)$ is a convex domain. In this article, we consider the Taylor coefficients $a_{n}(f)$ of the normalized expansion

$f(z)=z+n=2∑∞ a_{n}(f)z_{n},z∈D,$

and we impose on such functions $f$ the second normalization $f(1)=∞$. We call these functions concave schlicht functions, as the image of $D$ is a concave domain. We prove that the sharp inequalities

$∣a_{n}(f)−2n+1 ∣≤2n−1 ,n≥2,$

are valid. This settles a conjecture formulated in [2].

## Cite this article

Farit G. Avkhadiev, Christian Pommerenke, Karl-Joachim Wirths, Sharp inequalities for the coefficients of concave schlicht functions. Comment. Math. Helv. 81 (2006), no. 4, pp. 801–807

DOI 10.4171/CMH/74