Neighborhoods of analytic functions
type: Item
instance of: scholarly article
title: Neighborhoods of analytic functions
zbMATH Open document ID: 0664.30009
DOI: 10.1007/BF02820479
author: T. Sheil-Small
author: E. M. Silvia
published in: Journal d'Analyse Mathématique
publication date: 1989-01-01T00:00:00Z
review text: As a generalization of the definition of a \(\delta\)-neighbourhood of a function \[ f(z)=z+\sum^{\infty}_{k=2}a_ kz^ k \] holomorphic in the unit disc D given by \textit{St. Ruscheweyh} [Proc. Am. Math. Soc. 81, 521-527 (1981; Zbl 0458.30008)] the authors define T-\(\delta\)- neighbourhoods \(TN_{\delta}(f)\) of f as follows: Let \(T=\{T_ k\}^{\infty}_{k=2}\) be a sequence of nonnegative reals and A the set of functions \[ g(z)=z+\sum^{\infty}_{k=2}b_ kz^ k \] holomorphic in D. Then \[ TN_{\delta}(f)=\{g| \quad g\in A\quad and\quad \sum^{\infty}_{k=2}T_ k| a_ k-b_ k| \leq \delta \}. \] Sections 2-4 of this paper contain such a lot of interesting results on such neighborhoods concerning special subclasses of A that it is impossible to give an impression in few words. Anyone who is interested in Hadamard convolution should study this paper. The paper concludes with the discussion of two different neighbourhood systems and some open problems.
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C50
zbMATH DE Number: 4085985
zbMATH Keywords: Hadamard convolution
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02820479
OpenAlex ID: W1996776448
description: scientific article; zbMATH DE number 4085985