An integral formula for Taylor coefficients of a class of analytic functions
type: Item
instance of: scholarly article
title: An integral formula for Taylor coefficients of a class of analytic functions
zbMATH Open document ID: 0923.30003
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-05-24T00:00:00Z
review text: Let \(f(z)\) be a function analytic in \(\mathbb{C}\setminus[1,\infty)\) and let \(\Delta f(t) \lim_{\varepsilon\to 0^+} (f(t+i\varepsilon) -f(t-i \varepsilon))\). In this paper the following result is proved: Theorem. Let \(f\) be analytic in \(\mathbb{C}\setminus[1,\infty)\) and \(a_n\) be its Taylor coefficients. If \(f\) is bounded along \((1,\infty)\) such that \(| f(z)|= \circ (| z|^k)\) for some \(k\) and that \(\lim_{z\to 1}(1-z)f(z)=0\) and if \(\Delta f(x)\) exists for every \(x>1\), then for \(a_n\) the following integral representation \(a_n={1\over 2\Pi i} \int^\infty_1 {\Delta f(t) \over t^{n+1}} dt\) holds. Some applications of this theorem are also presented.
reviewed by: Jan Stankiewicz
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 30B10
Mathematics Subject Classification ID: 30E20
zbMATH DE Number: 1289961
zbMATH Keywords: integral formula
zbMATH Keywords: Taylor coefficients
author: Titus Hilberdink
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6293
OpenAlex ID: W2016501775
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DOI: 10.1006/JMAA.1999.6293
description: scientific article; zbMATH DE number 1289961