Operator $m$-convex functions

Abstract

The aim of this paper is to present a comprehensive study of operator $m$-convex functions‎.<br><br>‎Let $m\\in[0,1]$ and $J=[0,b]$ for some $b\\in\\mathbb{R}$ or‎ ‎$J=[0,\\infty)$‎. ‎A continuous‎<br><br>‎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.<br><br>‎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<br><br>‎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<br><br>‎$m$-Jensen operator functional and obtain an upper bound for it.