New Jensens Type Inequalities for Di¤erentiable Log-convex Functions of Selfadjoint Operators in Hilbert Spaces

Some new Jensen’s type inequalities for di¤erentiable log-convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications for particular cases of interest are also provided. 1. Introduction Jensen’s inequality for convex functions is one of the most important result in the Theory of Inequalities due to the fact that many other famous inequalities are particular cases of this for appropriate choices of the function involved, see for instance [9, p.]. The following result that provides an operator version for the Jensen inequality for convex functions is due to Mond and Peμcaríc [10] (see also [5, p. 5]): Theorem 1 (Mond-Peμcaríc, 1993, [10]). Let A be a selfadjoint operator on the Hilbert space H and assume that Sp (A) [m;M ] for some scalars m;M with m < M: If f is a convex function on [m;M ] ; then (MP) f (hAx; xi) hf (A)x; xi for each x 2 H with kxk = 1: Taking into account the above result and its applications for various concrete examples of convex functions, it is therefore natural to investigate the corresponding results for the case of log-convex functions, namely functions f : I ! (0;1) for which ln f is convex. We observe that such functions satisfy the elementary inequality f ((1 t) a+ tb) [f (a)] t [f (b)] for any a; b 2 I and t 2 [0; 1] : Also, due to the fact that the weighted geometric mean is less than the weighted arithmetic mean, it follows that any log-convex function is a convex functions. However, obviously, there are functions that are convex but not log-convex. As an immediate consequence of the Mond-Peμcaríc inequality above we can provide the following result, see for instance [4]: 1991 Mathematics Subject Classi…cation. 47A63; 47A99.