Reverse Jensen Integral Inequalities for Operator Convex Functions in Terms of Fréchet Derivative

Let f : I → R be an operator convex function of class C1 (I ). If (At )t∈T is a bounded continuous field of selfadjoint operators in B (H) with spectra contained in I defined on a locally compact Hausdorff space T with a bounded Radon measure μ, such that T 1dμ(t) = 1, then we obtain among others the following reverse of Jensen’s inequality: 0 ≤ T f (At ) dμ(t) − f T Asdμ(s) ≤ T Df (At ) (At ) dμ(t) − T Df (At ) T Asdμ(s) dμ(t) in terms of the Fréchet derivative Df (·)(·). Some applications for the Hermite– Hadamard inequalities are also given.