Regular growth of various characteristics of entire functions of order zero
type: Item
instance of: scholarly article
title: Regular growth of various characteristics of entire functions of order zero
zbMATH Open document ID: 1362.30039
DOI: 10.1134/S0001434616090042
author: N. V. Zabolotskii
author: O. V. Kostyuk
published in: Mathematical Notes
publication date: 2017-02-09T00:00:00Z
review text: This article is a continuation of the research of the first author which extended the theory of B. Ya. Levin and A. Pflüger of entire functions of completely regular growth of positive order to entire functions of order zero. In [Math. Notes 63, No. 2, 196--208 (1998; Zbl 0915.30026); translation from Mat. Zametki 63, No. 2, 172--182 (1998)], the first author introduced the notion of strongly regular growth for entire functions \(f\) of order zero. There the author considered the function \(r^{\lambda(r)}\), where \(\lambda(r)\) is the zero proximate order of the counting function \(n(r)\) of the zeros of \(f\) such that \(r^{\lambda(r)}\uparrow+\infty\) as \(r\to+\infty\). It was shown that the existence of the angular density of the zeros of an entire function \(f\) of order zero with respect to the function \(r^{\lambda(r)}\) implies the strongly regular growth of \(f\), and if the zeros of \(f\) are located on a finite system of rays, then also the converse statement is valid. In the general case, the converse statement does not hold. Sufficient conditions under which the strongly regular growth of an entire function of order zero implies the existence of the angular density of its zeros were obtained by the first author [Math. Notes 73, No. 5, 656--661 (2003; Zbl 1126.30021); translation from Mat. Zametki 73, No. 5, 698--703 (2003)]. In the present paper, the authors study the relationship between the strongly regular growth of an entire function \(f\) of zero order, the existence of the angular density of its zeros, the behavior of the Fourier coefficients of \(\log f\), and the regular growth of \(\log|f|\) and the argument of \(f\) in the \(L_p[0; 2\pi]\)-metric, \(p\geq1\). Let \(L\) be the class of continuously differentiable growth functions \(v\) on \(\mathbb{R}_+\) for which \(rv'(r)/v(r)\to0\) as \(r\to+\infty\). Let \(\mathcal{H}_0\) be the class of entire functions of order zero and let \(\mathcal{H}_0(v)\subset\mathcal{H}_0\) be the class of functio…
Mathematics Subject Classification ID: 30D15
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 6682247
zbMATH Keywords: entire functions
zbMATH Keywords: angular density
zbMATH Keywords: order of an entire function
zbMATH Keywords: strongly regular growth
MaRDI profile type: Publication
cites work: Q3848344
cites work: Q3874480
cites work: Q3766032
cites work: Concentration index of a subharmomic function of zeroth order
cites work: Strongly regular growth of entire functions of order zero
cites work: An example of entire function of strongly regular growth
cites work: Existence of an angular density of roots of zero-order entire functions.
cites work: Regularly varying functions
cites work: Q4899585
cites work: Criteria for the regularity of growth of the logarithm of modulus and the argument of an entire function
cites work: Q3376821
cites work: Asymptotic behavior of the logarithmic derivatives and the logarithms of meromorphic functions of completely regular growth. I
cites work: Q4749793
description: scientific article; zbMATH DE number 6682247