Matrix inequalities and majorizations around Hermite-Hadamard's inequality

We study the classical Hermite-Hadamard inequality in the matrix setting. This leads to a number of interesting matrix inequalities such as the Schatten p-norm estimates (parallel to A(q)parallel to(p)(p) + parallel to B-q parallel to(p)(p))(1/p) <= parallel to (xA + (1-xB)(q)parallel to(p) + parallel to ((1-x)A+xB)(q)parallel to(p), for all positive (semidefinite) n x n matrices A, B and y < q, x < O. A related decomposition, with the assumption X* X + Y* Y = XX* + YY* = I, is (X*AX+Y*BY)circle plus(Y*AY+X*BX) = 1/2n Sigma U-n(k=12)k(A circle plus B)U*(k), for some family of on x o n unitary matrices U-k. This is a majorization which is obtained by using the Hansen-Pedersen trace inequality.