Singular value inequalities with applications

Let A(i), B-i, X-i, Y-i be n x n complex matrices, i = 1, 2, ..., in and let f be a nonnegative increasing convex function on an interval I such that 0 is an element of I and f(0) <= 0. Then 2sj(f(vertical bar Sigma(m)(i=1) A(i)X(i)Y(i)*B-i*vertical bar)) <= (max {S, T})(2) s(j)(k) for j = 1, 2, ..., n, where S = parallel to Sigma(m)(i=1) A(i)A(i)*parallel to(1/2), T = parallel to Sigma(m)(i=1) BiBi*parallel to(1/2), K = f(vertical bar X-1 vertical bar(2) + vertical bar Y-1 vertical bar(2)) circle plus ... circle plus f(vertical bar X-m vertical bar(2) + vertical bar Y-m vertical bar(2)) and max {S, T} <= 1. Several singular value inequalities are also proved.