In this paper, we give the Alzer inequality for Hilbert space operators as follows:Let A, B be two selfadjoint operators on a Hilbert space H such that 02 are arithmetic and geometric means of A, B, respectively, where 0 < λ < 1. We show that if A and B are commuting, thenwhere A ′ := I − A, B ′ := I − B and 0 < λ ≤ 1 2 . Also, we state an open problem for an extension of Alzer inequality.

No abstract

We investigate a notion of relative operator entropy, which develops the theory started by J. I. Fujii and E. Kamei [Math. Japonica 34 (1989), 341-348]. For two finite sequences A = (A1, . . . , An) and B = (B1, . . . , Bn) of positive operators acting on a Hilbert space, a real number q and an operator monotone function f we extend the concept of entropy by setting

In this paper, we study the Hadamard's inequality for midconvex and quasi-midconvex functions in topological groups. A mapping naturally connected with this inequality and related result is also pointed out.

We study some properties of -normal operators and we present various inequalities between the operator norm and the numerical radius of -normal operators on Banach algebraℬ() of all bounded linear operators , where is Hilbert space.