Sharp inequalities for the coefficients of concave schlicht functions
type: Item
instance of: scholarly article
title: Sharp inequalities for the coefficients of concave schlicht functions
zbMATH Open document ID: 1210.30005
DOI: 10.4171/CMH/74
published in: Commentarii Mathematici Helvetici
publication date: 2006-11-13T00:00:00Z
review text: Summary: 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 by the authors in [Math. Nachr. 271, 3--9 (2004; Zbl 1149.30013)].
Mathematics Subject Classification ID: 30C50
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 5071521
zbMATH Keywords: Taylor coefficients
zbMATH Keywords: concave schlicht functions
zbMATH Keywords: slit mappings
author: F. G. Avkhadiev
author: Christian Pommerenke
author: Karl-Joachim Wirths
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/cmh/74
OpenAlex ID: W2047971998
description: scientific article; zbMATH DE number 5071521