Operator m-convex functions

The aim of this paper is to present a comprehensive study of operator m-convex functions. Let m epsilon [0, 1], and J = [0, b] for some b epsilon R or J = [0, infinity). A continuous function phi: J -> R is called operator mconvex if for any t epsilon [0, 1] and any self-adjoint operators A, B epsilon B(H), whose spectra are contained in J, we have phi(tA + m(1 -t) B) <= t phi(A) + m(1 -t) phi(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. 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.