Operator inequalities of Jensen type

Abstract

We present some generalized Jensen type operator inequalities involving sequences of self-adjoint operators. Among other things, we prove that if $f:[0,\infty )\to \mathbb{R}$ is a continuous convex function with $f(0)\le 0$, then {equation*} \sum_{i=1}^{n} f(C_i) \leq f(\sum_{i=1}^{n}C_i)-δ_f\sum_{i=1}^{n}\widetilde{C}_i\leq f(\sum_{i=1}^{n}C_i) {equation*} for all operators $C_i$ such that $0\le C_i\le M\le {\sum }_{i=1}^n{C}_i$ \ $(i=1,...,n)$ for some scalar $M\ge 0$, where $\widetilde{C_i}=1/2-|\frac{C_i}{M}-1/2|$ and ${\delta }_f=f(0)+f(M)-2f(\frac{M}{2})$.

17 pages, to appear in Topological Algebra and its Applications (a newly established journal by Versita Ltd.))