On discrete inequalities of the Poincaré type
type: Item
instance of: scholarly article
title: On discrete inequalities of the Poincaré type
zbMATH Open document ID: 0666.26010
DOI: 10.1007/BF01850291
author: B. G. Pachpatte
published in: Periodica Mathematica Hungarica
publication date: 1988-01-01T00:00:00Z
review text: The author presents four inequalities involving functions of many variables \(u:\quad N^ n\to {\mathbb{R}}\) \((N=\{1,2,...\}\), \(n\geq 2)\) and such that \(u(y_ 1,...,y_ n)=0\) if \(y_ i=1\) or \(y_ i=a_ i\) for some integer \(i\in \{1,...,n\}\). The point \(\bar a=(a_ 1,...,a_ n)\) is given. For example the author proves the inequality \[ \sum^{\bar a}_{\bar y=\bar 1}\{\prod^{r}_{m=1}| u_ m(\bar y)|^{p_ m}\}^{1/r}\leq \frac{1}{nr}\{\frac{a}{2}\}^{p/r}\sum^{r}_{m=1}\{\sum^{\bar a}_{\bar y=\bar 1}| Du_ m(\bar y)|^{p_ m}\}, \] where \(p=\sum^{r}_{m=1}p_ m\), \(p_ m\geq 2\), \(a=\max \{a_ 1,...,a_ n\}\), \[ \sum^{\bar a}_{\bar y=\bar 1}c(\bar y)=\sum^{a_ 1}_{y_ 1=1}...\sum^{a_ n}_{y_ n=1}c(y_ 1,...,y_ n),\quad | Dc(\bar y)| =\{\sum^{n}_{i=1}| \Delta_ ic(\bar y)|^ 2\}^{1/2}, \] and \[ \Delta_ ic(\bar y)=c(y_ 1,...,y_{i-1},y_ i+1,y_{i+1},...,y_ n)-c(y_ 1,...,y_{i- 1},y_ i,...,y_ n). \]
reviewed by: Jerzy Ryszard Popenda
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 39A12
zbMATH DE Number: 4089832
zbMATH Keywords: Poincaré type discrete inequalities
zbMATH Keywords: forward differences
zbMATH Keywords: integral inequalities
MaRDI profile type: Publication
cites work: Q3905923
cites work: Discrete analogs of inequalities of Wirtinger
cites work: Integral bounds for solutions of nonlinear reaction-diffusion equations
cites work: On the domain of attraction for steady states in heat conduction
cites work: Q3260749
cites work: On Poincaré-type integral inequalities
cites work: On Sobolev type integral inequalities
full work available at URL: https://doi.org/10.1007/bf01850291
OpenAlex ID: W1994408451
description: scientific article; zbMATH DE number 4089832