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

Let f:I -> R be an operator convex function of class C-1(I). If (A(t))(t is an element of 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 mu, such that integral(T)1d mu(t) = 1, then we obtain among others the following reverse of Jensen's inequality: 0 <= integral(T)f(A(t))d(mu)(t) - f(integral(T)A(s)d(mu)(s)) <= integral D-T(f)(A(t))d(mu)(t) - integral(T)Df(A(t))(integral(T)A(s)d(mu)(s))d mu(t) in terms of the Frechet derivativeDf(center dot)(center dot). Some applications for the Hermite-Hadamard inequalities are also given.