A characterization of the exponential function by product
type: Item
instance of: scholarly article
title: A characterization of the exponential function by product
zbMATH Open document ID: 0556.30022
author: Shigeru Kimura
published in: Kodai Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: This paper concerns mostly entire functions f(z) of positive integral order p. Conditions are used, implying \(f(z)=e^{P(z)}\), where P(z) is a polynomial of degree p. The main condition is (let \(\omega =\exp (\pi i/p))\) that \(f(z)f(\omega z)...f(\omega^{2p-1}z)\) is bounded as \(z\to \infty\) along some Jordan curve. Especially, for \(p=1\), \(f(z)=Ae^{Bz}\) if \(f(r)f(-r)\) is bounded and - as extra condition - all zeros are negative. Other extra conditions used are, e.g., zero-free sectors, zero being a defective value, only real or only non-real zeros. A skilful use of integral inequalities is made.
Mathematics Subject Classification ID: 30D20
zbMATH DE Number: 3886230
zbMATH Keywords: bounded products
reviewed by: Bo Kjellberg
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138036853
OpenAlex ID: W4246114272
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DOI: 10.2996/KMJ/1138036853
description: scientific article; zbMATH DE number 3886230