Inequalities for residuals of power expansions for the exponential function and completely monotone functions
type: Item
instance of: scholarly article
title: Inequalities for residuals of power expansions for the exponential function and completely monotone functions
zbMATH Open document ID: 0886.26014
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-05-10T00:00:00Z
review text: Let \(I_n(x)= \sum^{+\infty}_{k=n+1} (-1)^kx^k/k!\) \((x>0)\), \(n=1,2,\dots\) . Put \[ J_n(x):= I_{n-1}(x) I_{n+1}(x)/I^2_n(x). \] The author proves that: \[ n/(n+1)\leq J_n(x)\leq (n+1)/(n+ 2)\tag{1} \] for all \(x>0\), \(n\geq 1\); and that \(\lim_{x\to 0} J_n(x)= (n+1)/(n+2)\), \(\lim_{x\to\infty} J_n(x)= n/(n+ 1)\); \[ |I_n(x)|= x^{n+1}/(n+ 1)!(1+ x/(n+ a(x))),\tag{2} \] where \(a(x)\in (1,2)\) \((x>0)\) and \(\lim_{x\to 0} a(x)= 2\), \(\lim_{x\to\infty} a(x)= 1\); (3) The left-side inequality of (1) is valid (in some sense) for a general completely monotonic function \(f:(0,\infty)\to\mathbb{R}\). The proof is based essentially on Maclaurin's expansion with integral remainder term, as well as certain properties of the confluent hypergeometric function.
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26D07
Mathematics Subject Classification ID: 33C15
zbMATH DE Number: 1053555
zbMATH Keywords: inequalities for the exponential function
zbMATH Keywords: completely monotonic function
zbMATH Keywords: confluent hypergeometric function
author: Milan Merkle
reviewed by: József Sándor
MaRDI profile type: Publication
OpenAlex ID: W2094745961
cites work: Q3661894
cites work: An inequality for the exponential function
cites work: Inequalities for the tails of some elementary series
cites work: Q5841724
cites work: Notes on an inequality for sections of certain power series
cites work: Q5842462
cites work: An inequality for section of certain power series
cites work: Zeros of sections of power series
cites work: Kolmogorov-Landau inequalities for monotone functions
cites work: Q5721282
cites work: Q5788203
cites work: Q5621398
cites work: Q4089122
cites work: Q5844644
cites work: Q3990208
cites work: Some inequalities for the Chi square distribution function and the exponential functions
cites work: Q4864276
cites work: An inequality for residual of Maclaurin expansion
cites work: Analytic Inequalities
cites work: Q5772238
DOI: 10.1006/JMAA.1997.5485
description: scientific article; zbMATH DE number 1053555