A survey of Jensen type inequalities for log-convex functions of selfadjoint operators in Hilbert spaces
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)].
scientific article; zbMATH DE number 6008772