Some inequalities for unitarily invariant norms

We shall prove the inequalities vertical bar vertical bar vertical bar(A+B)(A+B)*vertical bar vertical bar vertical bar <= vertical bar vertical bar vertical bar AA* + BB* + 2AB*vertical bar vertical bar vertical bar <= vertical bar vertical bar vertical bar(A-B)(A-B)* + 4AB*vertical bar vertical bar vertical bar for all n x n complex matrices A, B and all unitarily invariant norms vertical bar vertical bar vertical bar.vertical bar vertical bar vertical bar If further A, B are positive definite it is proved that Pi(k)(j=1) lambda j(A(alpha)B) <= Pi(k)(j=1) lambda(j)(A(1-alpha)B(alpha)), 1 <= k <= n, 0 <= alpha <= 1 where (sic)(alpha) denotes the operator means considered by Kubo and Ando and lambda j(X), 1 <= j <= n, denote the eigenvalues of X arranged in the decreasing order whenever these all are real. A number of inequalities are obtained as applications. (C) 2010 Elsevier Inc. All rights reserved.