Submajorization inequalities for τ - measurable operators

Let (M, tau) be a semi-finite von Neumann algebra and f be a nonnegative function on [0, infinity) with f (0) = 0. Let x1, . . . ,x(n) be tau-measurable operators and let alpha(1), . . . alpha(n) be positive real numbers such that Sigma(j =1) (n) alpha(j) = 1. We have the following results. 1. If g(t) = f root t is convex, then f(broken vertical bar Sigma(n)(j=1) alpha(j)x(j)broken vertical bar + Sigma(1 <= j <= k <= n) f( root alpha j alpha k broken vertical bar xj - xk broken vertical bar less than or similar to Sigma(n)(j=1) alpha jf(broken vertical bar x(j))broken vertical bar) 2. If h(t) = f(root t) is concave, then Sigma(n)(j=1) alpha(j) f(broken vertical bar x(j)broken vertical bar less than or similar to f(broken vertical bar Sigma(n)(j=1) alpha(j)x(j) broken vertical bar) + Sigma(1 <= j <= k <= n) f(root alpha(j)alpha(k) broken vertical bar x(j) - x(k)broken vertical bar).