Singular values of differences of positive semidefinite matrices

Let M-n be the space of n x n complex matrices. For A is an element of M-n, let s (A) = (s(1) (A),..., s(n)(A)), where s(1)(A) greater than or equal to ... greater than or equal to s(n)(A) are the singular values of A. We prove that if A, B is an element of M-n are positive semidefinite, then (i) s(j) (A - B) less than or equal to s(j) (A+B), j = 1, 2,..., n, and (ii) the weak log-majorization relations s (A-\z\B) <(wlog) s (A + zB) <(wlog) s (A + \z\B) hold for any complex number z. This sharpens some results due to R. Bhatia and F. Kittaneh.