Some inequalities for eigenvalues and symplectic eigenvalues of positive definite matrices

We show that for any two n x n matrices X and Y we have the inequality s(j)(2) (I + XY) <= lambda(j) ((I+X * X)(I+Y * Y)), where s(j)(T) and lambda(j) (T) denote the decreasingly ordered singular values and eigenvalues of T. As an application, we show that for 2n x 2n real positive definite matrices the symplectic eigenvalues d(j), under some special conditions, satisfy the inequality d(j) (A + B) >= d(j)(A) + d(1)(B).