A survey of Jensen type inequalities for log-convex functions of selfadjoint operators in Hilbert spaces
type: Item
instance of: scholarly article
title: A survey of Jensen type inequalities for log-convex functions of selfadjoint operators in Hilbert spaces
zbMATH Open document ID: 1242.47015
published in: Communications in Mathematical Analysis
publication date: 2012-02-22T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.cma/1305810737
review text: This paper is a survey article of Jensen type inequalities for log-convex functions of selfadjoint operators on a Hilbert space. Almost all the results are from the author's own two papers ([``Some Jensen's type inequalities for log-convex functions of selfadjoint operators in Hilbert spaces'', Preprint RGMIA Res. Rep. Coll., 13(e), Art. 2 (2010), cf. Bull. Malays. Math. Sci. Soc. (2) 34, No.~3, 445--454 (2011; Zbl 1239.47014)], [``New Jensen's type inequalities for diffrentiable log-convex functions of selfadjoint operators in Hilbert spaces'', Preprint RGMIA Res. Rep. Coll., 13(e), Art. 3 (2010)]). The statements of the theorems are rather lengthy. The proofs are based mainly on two inequalities, which the author calls (MP) inequality and (P) inequality. The first inequality, \(f(\langle Ax,x\rangle)\leq \langle f(A)x,x\rangle\), where \(f\) is a convex function defined on an interval containing the spectrum of the selfadjoint operator \(A\) and \(x\) is a unit vector, is essentially due to \textit{J. L. W. V. Jensen} [Nyt. Tidss. for Math. 16B, 49--68 (1905; JFM 36.0447.03)] (in the case of matrices), though not written specifically in this form. The author calls this the Mond-Pečarić (MP) inequality after [\textit{B. Mond} and \textit{J. E. Pečarić}, Houston J. Math. 19, No. 3, 405--420 (1993; Zbl 0813.46016)].
reviewed by: Jaspal Singh Aujla
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 47-02
zbMATH DE Number: 6008772
zbMATH Keywords: selfadjoint opertaors
zbMATH Keywords: log-convex functions
zbMATH Keywords: Jensen's inequality
author: Silvestru Sever Dragomir
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6008772