## A survey of Jensen type inequalities for log-convex functions of selfadjoint operators in Hilbert spaces.(English)Zbl 1242.47015

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 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 [B. Mond and J. E. Pečarić, Houston J. Math. 19, No. 3, 405–420 (1993; Zbl 0813.46016)].

### MSC:

 47A63 Linear operator inequalities 47-02 Research exposition (monographs, survey articles) pertaining to operator theory

### Citations:

Zbl 0813.46016; Zbl 1239.47014; JFM 36.0447.03
Full Text: