Uniform estimates of entire functions by logarithmic sums
type: Item
instance of: scholarly article
title: Uniform estimates of entire functions by logarithmic sums
zbMATH Open document ID: 0886.30022
published in: Journal of Functional Analysis
publication date: 1998-04-26T00:00:00Z
review text: An entire function \(f\) of exponential type is said to be of the Cartwright class if \(\int\frac{\log^+|f(t)|}{t^2+1}dt<+\infty\). For \(B>0\) let \(\mathcal A(B)\) denote the set of all even entire functions \(f\) of the Cartwright class such that the exponential type of \(f\) is \(\leq B\), \(f|_{\mathbb R}\geq1\), and \(f(0)=1\). Define \(\mathcal I(f):=\int_0^\infty\frac{\log f(t)}{t^2}dt\), \(\mathcal S(f):=\sum_{n=1}^\infty\frac{\log f(n)}{n^2}\). The main result of the paper is the following theorem. There exists a constant \(T_\ast>0\) (\(T_\ast\simeq0.4422\)) such that if \(B<2T_\ast\), \((f_{k})_{k=1}\subset\mathcal A(B)\), and \(\mathcal S(f_{k})\longrightarrow0\), then \(\mathcal I(f_{k})\longrightarrow0\). Using the above theorem, the author proves the following two important results. (1) If \(f\) is an entire function of exponential type \(<T_\ast\) and \(\sum\frac{\log^+|f(n)|}{n^2+1}<+\infty\), then \(f\) is of the Cartwright class. (2) (A generalization of a theorem by Koosis.) \(\forall_{B_0<T_\ast}\), \(\forall_{\gamma>0}\) \(\exists_{\eta_0>0}\), \(\forall_{\eta\leq\eta_0}\) \(\exists_{C_\eta>0}\): if \(f\) is an entire function of exponential type \(\leq B\leq B_0\) with \(\sum\frac{\log^+|f(n)|}{n^2+1}\leq\eta\), then \(|f(z)|\leq C_\eta\exp(B|y|+\gamma|z|)\) for any \(z\in\mathbb C\).
Mathematics Subject Classification ID: 30D15
zbMATH DE Number: 1077108
author: Henrik L. Pedersen
reviewed by: Marek Jarnicki
MaRDI profile type: Publication
full work available at URL: jfan.1996.3046
OpenAlex ID: W2056251586
cites work: On Fourier transforms of measures with compact support
cites work: On the closure of characters and the zeros of entire functions
cites work: Q5833361
cites work: Linear and complex analysis problem book 3. Part 2
cites work: Weighted polynomial approximation on arithmetic progressions of intervals or points
cites work: Harmonic estimation in certain slit regions and a theorem of Beurling and Malliavin
cites work: Q3994552
cites work: Q3998719
cites work: Q3483772
cites work: A relation between two results about entire functions of exponential type
cites work: Q4866448
cites work: Q5531193
cites work: Sur la croissance radiale d'une fonction méromorphe
cites work: On the multiplier theorem for Fourier transforms of measures with compact support
cites work: Q4482959
DOI: 10.1006/JFAN.1996.3046
description: scientific article; zbMATH DE number 1077108