Notes on the Duren-Leung conjecture
type: Item
instance of: scholarly article
title: Notes on the Duren-Leung conjecture
zbMATH Open document ID: 1125.30012
author: Jian-Lin Li
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-05-23T00:00:00Z
review text: Assume that \(f(z)= z+ a_2z^2+\cdots\) is in the class \(S\) and \(\log{f(z)\over z}= 2\sum^\infty_{k=1}\gamma_k z^k\). Let \(\delta\) denote the Milin constant. The Duren-Leung conjecture asserts that \[ \sum^n_{k=1} |\gamma_k|\leq \sum^n_{k=1} k^{-2}\tag{\(*\)} \] and it remains open for \(n\geq 3\). The main objective of this note is to add some new observations concerning \((*)\). Main results of this paper are as follows: (1) if \(|a_2|\leq (4-75\delta/26)^{1/2}= 1.76\dots\) then \((*)\) holds for \(n\geq 3\), (2) if \(f(z)\) is not the Koebe function then there exists a constant \(N(f)\) such that \((*)\) holds for all \(n\geq N(f)\).
Mathematics Subject Classification ID: 30C50
zbMATH DE Number: 5156414
zbMATH Keywords: logarithmic coefficients
zbMATH Keywords: de Branges' theorem
zbMATH Keywords: Milin constant
reviewed by: Eligiusz Złotkiewicz
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.10.030
OpenAlex ID: W2039345981
Wikidata QID: Q123146265
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DOI: 10.1016/J.JMAA.2006.10.030
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description: scientific article; zbMATH DE number 5156414