Some inequalities for eigenvalues and symplectic eigenvalues of positive definite matrices

We show that for any two [Formula: see text] matrices [Formula: see text] and [Formula: see text] we have the inequality [Formula: see text] where [Formula: see text] and [Formula: see text] denote the decreasingly ordered singular values and eigenvalues of [Formula: see text]. As an application, we show that for [Formula: see text] real positive definite matrices the symplectic eigenvalues [Formula: see text] under some special conditions, satisfy the inequality [Formula: see text].