On the Bombieri numbers for the class S
type: Item
instance of: scholarly article
title: On the Bombieri numbers for the class S
zbMATH Open document ID: 1383.30008
author: Yuk-J Leung
published in: The Journal of Analysis
publication date: 2017-05-04T00:00:00Z
review text: Let be \[ f(z)=z+a_2 z^2+a_3 z^3 +\cdots + a_n z^n +\cdots \] analytic and univalent in the unit disk, then the Bombieri numbers are defined in the following way \[ \sigma_{mn} := \liminf_{a_m \to m} \frac{n - Re ( a_n )}{m - Re ( a_m )}. \] The question posed by \textit{E. Bombieri} [Invent. Math. 4, 26--67 (1967; Zbl 0174.12301)] is: are the Bombieri numbers equal to the numbers \[ B_{mn} := \min_{\theta} \frac{n \sin \theta -\sin {n\theta}}{m\sin \theta -\sin {m\theta}}. \] The aim of the paper is to show that this is wrong in many cases, more precisely it is proven that \(\sigma_{m2}< B_{m2} \) for all \(m>2\) and \(\sigma_{m3} <B_{m3} \) for all odd \(m>3\). Furthermore a new derivation of the second variation formula of Bombieri on the coefficient \(a_n\) near the Koebe function is given.
reviewed by: Bodo Dittmar
Mathematics Subject Classification ID: 30C75
Mathematics Subject Classification ID: 33C05
zbMATH DE Number: 6711589
zbMATH Keywords: Loewner chain
zbMATH Keywords: Bombieri numbers
MaRDI profile type: Publication
full work available at URL: s41478-016-0017-2
OpenAlex ID: W2556581841
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DOI: 10.1007/S41478-016-0017-2
description: scientific article; zbMATH DE number 6711589