Note on certain inequality
type: Item
instance of: scholarly article
title: Note on certain inequality
zbMATH Open document ID: 0895.26009
author: Ivica Gusić
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-02-09T00:00:00Z
review text: Let \(G\), \(H\) be lattice ordered groups, \(|\cdot|\) the function defined by \(| x|=\sup\{x, 0\}-\inf\{x, 0\}\) and \(f:G^+\to H^+\) a function with increasing increments, such that \(f(0)= 0\). The author presents a proof of the inequality \[ \sum^n_{k= 1}| f(a_k)- f(a_{k- 1})|\leq f\Biggl(\sum^n_{k= 1}| a_k- a_{k-1}|\Biggr), \] where \(0= a_0,a_1,\dots, a_n\in G^+\). Furthermore, he proves that if \(f\) is a function with strictly increasing increments, then the equality holds if and only if \(a_1\leq\cdots\leq a_n\). the above result is an interesting generalization of the result of \textit{L. Maligranda} and \textit{W. Orlicz} [Monatsh. Math. 104, No. 1, 53-65 (1987; Zbl 0623.26009)] and \textit{J. E. Pečarić} and the author [Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 36, 169-178 (1996)]. The author also proves a similar inequality for an increasing function \(f\) with decreasing increments and for comparable \(a_1,\dots, a_n\). Perhaps, the readers will find this article stimulating for consideration of similar problems on lattice ordered groups.
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 1115493
zbMATH Keywords: function with increasing increments
zbMATH Keywords: inequality
zbMATH Keywords: lattice ordered groups
reviewed by: J. Pečarić
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5686
OpenAlex ID: W2058459374
cites work: “N. Bourbaki”
cites work: On some propertis of functions of generalized variation
cites work: Q4337877
DOI: 10.1006/JMAA.1997.5686
description: scientific article; zbMATH DE number 1115493