Sharp bounds for integral means
type: Item
instance of: scholarly article
title: Sharp bounds for integral means
zbMATH Open document ID: 1011.26015
DOI: 10.1016/S0022-247X(02)00255-X
author: Minkyun Kim
author: C. J. Neugebauer
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: A sample of the results obtained here is the following theorem: Let \((X,{\mathcal M},m)\) be a measure space, and let \(E\in{\mathcal M}\). A number \(M\) is an \((a,A)\)-bound for \(f: E\to\mathbb{R}\) if there exist functions \(f_1,f_2: E\to\mathbb{R}\) and numbers \(a\leq A\) such that \[ M= A-a,\;f(x)= f_1(x)- f_2(x);\tag{1} \] \[ a\leq f_1(x), f_2(x)\leq A,\tag{2} \] \[ m(E) \int_e f_1f_2 dm\geq \int_E f_1dm \int_E f_2 dm.\tag{3} \] If \(M\) is an \((a,A)\)-bound for \(f\), and \(\int_E f dm= 0\), then \(\int_E|f|dm\leq {M\over 2} m(E)\), where \(m(E)\) denotes the measure of \(E\). Then sharp bounds are deduced for the average of \(|f|^p\) over \(E\). A new proof for the famous Grüss inequality is deduced. A new estimate for the error term in the trapezoidal rule is also given.
reviewed by: József Sándor
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 65D30
zbMATH DE Number: 1866703
zbMATH Keywords: integral means
zbMATH Keywords: sharp bounds
zbMATH Keywords: Grüss inequality
zbMATH Keywords: trapezoidal rule
MaRDI profile type: Publication
cites work: A Grüss type discrete inequality in inner product spaces and applications
cites work: A generalization of the Ostrowski integral inequality for mappings whose derivatives belong to \(L_p[a,b]\) and applications in numerical integration
full work available at URL: s0022-247x(02)00255-x
OpenAlex ID: W1995679197
description: scientific article; zbMATH DE number 1866703