Title : ( Operator $m$-convex functions )
Authors: J. Rooin , Mohammad Sal Moslehian ,Access to full-text not allowed by authors
Abstract
The aim of this paper is to present a comprehensive study of operator $m$-convex functions. Let $m\in[0,1]$ and $J=[0,b]$ for some $b\in\mathbb{R}$ or $J=[0,\infty)$. A continuous function $\varphi:J\to\mathbb{R}$ is called operator $m$-convex if for any $t\in[0,1]$ and any self-adjoint operators $A, B\in \mathbb{B}({\mathscr{H}})$, whose spectra are contained in $J$, $\varphi\big(tA+m(1-t)B\big)\leq t\varphi(A)+m(1-t)\varphi(B)$. We first generalize the celebrated Jensen inequality for continuous $m$-convex functions and Hilbert space operators and then use suitable weight functions to give some weighted refinements of it. Introducing the notion of operator $m$-convexity, we extend the Choi--Davis--Jensen inequality for operator $m$-convex functions.We also present an operator version of the Jensen--Mercer inequality for $m$-convex functions and generalize this inequality for operator $m$-convex functions involving continuous fields of operators and unital fields of positive linear mappings. Employing the Jensen--Mercer operator inequality for operator $m$-convex functions, we construct the $m$-Jensen operator functional and obtain an upper bound for it.