On coefficient estimates and conjectures for the class \(\Sigma\)
type: Item
instance of: scholarly article
title: On coefficient estimates and conjectures for the class \(\Sigma\)
zbMATH Open document ID: 0549.30011
DOI: 10.1007/BF01174885
author: Glenn Schober
author: John K. Williams
published in: Mathematische Zeitschrift
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173447
review text: Let \(\Sigma\) denote the family of univalent functions \(f(z)=z+\sum^{\infty}_{n=0}b_ nz^{-n}\) in \(D=\{z: | z| >1\}.\) Using the basic lemma with constraint from the paper of \textit{W. E. Kirwan} and \textit{G. Schober}, Math. Z. 180, 19-40 (1982; Zbl 0498.30024) and some results of Jenkins, Garabedian and Schiffer and Kubota several very nice inequalities for combinations of the coefficients of the functions from \(\Sigma\) are determined. Especially the conjecture of Kirwan that \(Re\{nb_ 1-b_ n\}\leq n,\quad n\geq 2\) for \(f\in\Sigma \) is examined. The paper is ended by four conjectures connected with Kirwan's conjecture. - This is a very interesting paper.
Mathematics Subject Classification ID: 30C50
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C35
zbMATH DE Number: 3875626
MaRDI profile type: Publication
Wikidata QID: Q122906463
cites work: Q5767886
cites work: On the second variation for univalent functions
cites work: A coefficient inequality for schlicht functions
cites work: On Certain Coefficients of Univalent Functions. II
cites work: New inequalities from old ones
cites work: A coefficient inequality for certain meromorphic univalent functions
cites work: On the fourth coefficient of meromorphic univalent functions
cites work: Low order coefficient estimates in the class \sigma
cites work: A Method of Variation Within the Family of Simple Functions
OpenAlex ID: W2056953758
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description: scientific article; zbMATH DE number 3875626