On some criteria for completely regular growth of entire functions of exponential type
type: Item
instance of: scholarly article
title: On some criteria for completely regular growth of entire functions of exponential type
zbMATH Open document ID: 1121.30014
author: V. B. Sherstyukov
published in: Mathematical Notes
publication date: 2007-05-21T00:00:00Z
review text: This paper investigates the relationship between the regularity of growth of an entire function of exponential type (EFET) \(\mathcal{L}(\lambda)\) with only simple zeros \(\lambda_k, \;k=1,2,\ldots.\) For EFET \(\mathcal{L}\) with real zeros the author obtains the growth regularity conditions in terms of the bounds from below for the absolute value of the derivative \(\mathcal{L^{\prime}}\) at points \(\lambda_k.\) These bounds have the form \[ \frac{1}{| {\mathcal{L}}^{\prime}(\lambda _k)| }=O(| \omega(\lambda_k)| ), \;k \to \infty, \] where \(\omega(\lambda)\) is an entire function of zero exponential type with some special properties. Further, the paper deals with the class \(L\) of EFET \(\mathcal{L}\) with only simple zeros \(\lambda _k, \;k=1,2,\ldots,\) satisfying the following condition \[ \limsup_{k \to \infty}\left\{ \frac{1}{| \lambda _k| }\ln \frac{1}{{\mathcal{L}}^{\prime}(| \lambda _k | ) }+h({\mathcal{L}}, \arg \lambda _k)\right\}\leq 0, \] where \(h(\mathcal{L},\theta)\) is the indicator of the function \({\mathcal{L}}, \;\theta \in [0; 2\pi ).\) Leont'ev posed the problem whether or not the function \({\mathcal{L}}\in L\) has a completely regular growth. This problem has not been solved. The author proves the following theorem. \textbf{Theorem.} The indicator diagram of an arbitrary function \({\mathcal{L}}\in L\) with only real zeros is a parallel to the imaginary axis segment . Moreover, if \(\mathcal{L}\) is an even function, then \(\mathcal{L}\) is a function of completely regular growth on the real axis and \( \lim _{k \to \infty}\frac{1}{| \lambda _k| }\ln \frac{1}{{\mathcal{L}}^{\prime}(| \lambda _k| ) }=0.\) Some sufficient conditions for a function \({\mathcal{L}}\in L\) with the positive indicator \(h({\mathcal{L}},\theta)\) to be a function of completely regular growth are obtained. For instance, it is proved that if the inverse to such function \(\mathcal{L}\) admits the representation \( \frac{1}{{\mathcal{L}}(\lambda)} =\sum_{k=1}^{\i…
reviewed by: Olga M. Katkova
Mathematics Subject Classification ID: 30D15
zbMATH DE Number: 5155503
zbMATH Keywords: entire function of exponential type
zbMATH Keywords: completely regular growth
zbMATH Keywords: exponential series
zbMATH Keywords: Leont'ev problem
zbMATH Keywords: entire function with simple real zeros
MaRDI profile type: Publication
full work available at URL: s11006-006-0115-6
OpenAlex ID: W1969787577
cites work: Q3848344
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DOI: 10.1007/S11006-006-0115-6
description: scientific article; zbMATH DE number 5155503