On some propertis of functions of generalized variation
type: Item
instance of: scholarly article
title: On some propertis of functions of generalized variation
zbMATH Open document ID: 0623.26009
DOI: 10.1007/BF01540525
published in: Monatshefte für Mathematik
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178339
review text: We show that if \(\phi\) is a convex \(\phi\)-function then the space \(V^*_{\phi}\), generated by a \(\phi\)-variation, forms a commutative Banach algebra with respect to the pointwise multiplication under the norms \[ \| x\| =\sup_{t\in [a,b]}| x(t)| +\| x\|^ v_{\phi}\quad or\quad \| x\| =2\phi^{-1}(1)\| x\|^ v_{\phi}, \] where \[ \| x\|^ v_{\phi}=\inf \{r>0: var_{\phi}(x/r):=\sup_{\pi}\sum^{n}_{k=1}\phi (| x(t_ k)- x(t_{k-1})| /r)\leq 1\}. \] However, if \(\phi (u)=u^ p\), \(0<p\leq 1\), then \(var_ p(xy)\leq var_ p(x)var_ p(y).\) Moreover, for any convex \(\phi\)-function \(\phi\) \(var(\phi (| x|))\leq \phi (var(| x|)).\) The above inequality is used in a new proof of the generalized Opial's inequality.
Mathematics Subject Classification ID: 26A45
Mathematics Subject Classification ID: 46J10
zbMATH DE Number: 4010825
zbMATH Keywords: functions of bounded generalized variation
zbMATH Keywords: commutative Banach algebra
zbMATH Keywords: generalized Opial's inequality
author: Lech Maligranda
author: Władysław Orlicz
MaRDI profile type: Publication
cites work: An Inequality for Variations
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OpenAlex ID: W2051069706
description: scientific article; zbMATH DE number 4010825