An estimate of quasi-arithmetic mean for convex functions
type: Item
instance of: scholarly article
zbMATH Open document ID: 1284.47016
author: Jadranka Mićić
author: Yuki Seo
author: J. Pečarić
publication date: 2013-07-12T00:00:00Z
full work available at URL: 2012.html
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 47A64
zbMATH DE Number: 6187920
zbMATH Keywords: quasi-arithmetic mean
zbMATH Keywords: positive linear map
zbMATH Keywords: positive operator
zbMATH Keywords: Jensen inequality
zbMATH Keywords: Mond-Pečarić method
MaRDI profile type: Publication
title: An estimate of quasi-arithmetic mean for convex functions
published in: Scientiae Mathematicae Japonicae
review text: For a selfadjoint operator \(A\) on a Hilbert space \(H\), a normalized positive linear map \(\Phi\) and a strictly monotone function \(\varphi\), a quasi-arithmetic mean is defined by \(\varphi^{-1}(\Phi(\varphi(A)))\). \textit{A. Matsumoto} and \textit{M. Tominaga} [Sci. Math. Jpn. 61, No. 2, 243--247 (2005; Zbl 1092.47018)] investigated the relation between \(\varphi^{-1}(\Phi(\varphi(A)))\) and \(\Phi(A)\) for a convex function \(\varphi\). In this paper, the authors provide some conditions under which an order among quasi-arithmetic mean of the form \(\varphi^{-1}(\Phi(\varphi(A)))\leq \psi^{-1}(\Phi(\psi(A)))\) holds. Some difference and ratio type complementary inequalities among quasi-arithmetic means are given by using the Mond-Pečarić method, see [\textit{J. Pečarić} et al., Mond-Pečarić method in operator inequalities. Inequalities for bounded selfadjoint operators on a Hilbert space. Monographs in Inequalities 1. Zagreb: ELEMENT (2005; Zbl 1135.47012)].
reviewed by: Mohammad Sal Moslehian
description: scientific article; zbMATH DE number 6187920