Some singular value inequalities on majorisation for positive semi-definite matrices

In this note, some singular value inequalities on majorisation for positive semi-definite matrices are given. First, the weak log-majorisation of the quantity f (A)g(B) +/- f (B)g(A) for certain functions is obtained, where A, B >= 0. As a consequence, the weak log-majorisation of the quantity A(p) B-q +/- B-p A(q) for p, q > 0 is also obtained. Finally, different estimates for the singular values of the quantity A(p) B-q +/- B-p A(q) are deduced for A, B > 0 and 0 < p <= q.