REFINEMENTS OF THE OPERATOR JENSEN-MERCER INEQUALITY

A Hermite-Hadamard-Mercer type inequality is presented and then generalized to Hilbert space operators. It is shown that f M+m-Sigma(n)(i=1) x(i)A(i)) <= f(M)+f(m)-Sigma(n)(i=1)f(x(i))A(i), where f is a convex function on an interval [m,M] containing 0, x(i) is an element of [m, M], i=1,...,n, and A(i) are positive operators acting on a finite dimensional Hilbert space whose sum is equal to the identity operator.A Jensen-Mercer operator type inequality for separately operator convex functions is also presented