Complete equivalence of the logarithms of the maximum modulus and the maximal term of an entire Dirichlet series
type: Item
instance of: scholarly article
title: Complete equivalence of the logarithms of the maximum modulus and the maximal term of an entire Dirichlet series
zbMATH Open document ID: 0709.30001
DOI: 10.1007/BF01170894
author: Myroslav Sheremeta
published in: Mathematical Notes
publication date: 1990-01-01T00:00:00Z
review text: Let \(\Lambda =(\lambda_ n)^{\infty}_{n=0}\), \(0\leq \lambda_ n\nearrow +\infty\) and let S(\(\Lambda\)) denote the set of all entire functions F of the form \(F(z)=\sum^{\infty}_{n=0}a_ n \exp (\lambda_ nz)\). Put \[ M(\sigma,F):=\sup \{| F(\sigma +it)|:\;t\in {\mathbb{R}}\},\quad \mu (\sigma,F):=\max \{| a_ n| \exp (\lambda_ n\sigma):\;n\geq 0\}. \] The author studies the problem of the asymptotic behaviour of the functions log M(\(\sigma\),F) and log \(\mu\) (\(\sigma\),F) when \(\sigma \to +\infty\). He proves that for any \(\Lambda\) there exists a function \(F\in S(\Lambda)\) for which the relation \[ (*)\quad \log M(\sigma,F)\sim \log \mu (\sigma,f),\quad \sigma \to +\infty, \] is not fulfilled. Moreover, he characterizes some classes of sequences \(\Lambda\) and functions \(F\in S(\Lambda)\) for which condition (*) is satisfied.
Mathematics Subject Classification ID: 30B50
Mathematics Subject Classification ID: 30D15
zbMATH DE Number: 4165234
zbMATH Keywords: Dirichlet series
reviewed by: Marek Jarnicki
MaRDI profile type: Publication
cites work: Q3684374
cites work: Q3772452
full work available at URL: https://doi.org/10.1007/bf01170894
OpenAlex ID: W2070694976
description: scientific article; zbMATH DE number 4165234