Submajorization inequalities of τ-measurable operators

Let M be a finite von Neumann algebra with a normal faithful finite trace tau. Let x, y be measurable positive operators with respect to (M, tau), and z is an element of M be an expansive operator. We prove f (z*xz) less than or similar to z*f(x)z for any concave function f : [0, infinity) -> [0, infinity), and z*g(x)z less than or similar to g(z*xz) for any convex function g : [0, infinity) -> [0, infinity) with g(0) = 0.