Symmetric norms and reverse inequalities to Davis and Hansen-Pedersen characterizations of operator convexity

Let A, B, Z be n-by-n matrices. Suppose AB >= 0 (positive semi-definite) and Z > 0 with extremal eigenvalues a and b. Then, the sharp inequality parallel to ZAB parallel to <= a + b/2 root ab parallel to BZA parallel to holds for every unitarily invariant norm. Among the consequences, we get the operator inequality XZX <= [(a + b)(2)/4ab]Z for every 0 <= X <= I. and some Kantorovich type inequalities (MondPecaric inequalities). Also in connection, reverse inequalities of Davis and Hansen-Pedersen characterizations of operator convexity are established. For instance, given any operator convex function f : [0, infinity) --> [0, infinity) and any subspace E, f(Z(E)) >= 4ab/(a+b)(2) (f(Z))(E). In passing, we point out a simplified proof of Hansen-Pedersen's inequality.