Coefficient estimates for the class \(\Sigma\)
type: Item
instance of: scholarly article
title: Coefficient estimates for the class \(\Sigma\)
zbMATH Open document ID: 0589.30018
author: Mitsuru Ozawa
published in: Kodai Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: Let \(\Sigma\) be the class of functions \(f(z)=z+\sum^{\infty}_{n=0}b_ nz^{-n}\) that are analytic and univalent for \(| z| >1\). For each \(n\geq 2\), \textit{Y. J. Leung} and the reviewer [Proc. Am. Math. Soc. 94, 659-664 (1985; Zbl 0584.30013)] proved that there is a finite value t, depending on n, such that the inequality \(Re\{tb_ 1+b_ n\}\leq t\) is valid for all functions in \(\Sigma\). As soon as this inequality is valid for one value of t, it is easy to see that it remains valid for all larger ones. Denote by \(A_ n\) the best (smallest) value for t. In this article the author shows that \(A_ 3\leq 2\), \(A_ 5\leq (27+8\sqrt{3})/12\), \(A_ 7\leq 5.5\), \(A_ 9<8\), and \(A_{11}<10\). The area theorem reduces the estimates to when \(b_ 1\) is near 1. Then the author makes profound use of Grunsky's inequalities. About the same time, Leung proved that \((e^ 4+3)/(e^ 4-1)\leq A_ 3\leq 2\) and conjectured that \(A_ 3=(e^ 4+3)/(e^ 4-1)\). Very recently, Ozawa has verified this conjecture.
Mathematics Subject Classification ID: 30C50
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 3946536
zbMATH Keywords: area theorem
zbMATH Keywords: Grunsky's inequalities
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138037155
OpenAlex ID: W2003679090
cites work: A coefficient inequality for schlicht functions
cites work: New inequalities from old ones
cites work: On coefficient estimates and conjectures for the class \(\Sigma\)
cites work: High Order Coefficient Estimates in the Class &#931
cites work: Low order coefficient estimates in the class \sigma
DOI: 10.2996/KMJ/1138037155
description: scientific article; zbMATH DE number 3946536