On the coefficients of small univalent functions
type: Item
instance of: scholarly article
title: On the coefficients of small univalent functions
zbMATH Open document ID: 0894.30012
DOI: 10.1007/BF03322071
published in: Results in Mathematics
publication date: 1998-08-30T00:00:00Z
review text: In this paper, a univalent function \(f(z)= \sum^\infty_{n=0} a_nz^n\) is small if \(\alpha>0\) is small in the inequality \((1-| z|^2) | f''(z)/f'(z) |\leq \alpha\) for \(| z | <1\). The authors prove the following theorem. For sufficiently small \(\alpha>0\), there exists a univalent function \(f\) with \((1-| z|^2) | f''(z)/f'(z) |\leq \alpha\) for \(| z | <1\) such that, for some constant \(c>0\), the inequality \(| a_n|> n^{c\alpha^2-1}\) holds for infinitely many \(n\). The proof depends on results by \textit{N. G. Makarov} [Proc. Am. Math. Soc. 96, 233-236 (1986; Zbl 0623.30025)] and by \textit{L. Carleson} and \textit{P. W. Jones} [Duke Math. J. 66, No. 2, 169-206 (1992; Zbl 0765.30005)].
reviewed by: Renate McLaughlin
Mathematics Subject Classification ID: 30C50
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 1140107
author: Martin Chuaqui
author: Brad G. Osgood
author: Christian Pommerenke
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03322071
OpenAlex ID: W1965973566
description: scientific article; zbMATH DE number 1140107