New Hilbert–Schmidt norm inequalities for positive semidefinite matrices

Let A and B be positive semidefinite matrices, and let X be any matrix. As a generalization of an earlier Hilbert–Schmidt norm inequality, we prove that A s X + X B 1 - s 2 2 + A 1 - s X + X B s 2 2 ≤ A t X + X B 1 - t 2 2 + A 1 - t X + X B t 2 2 for 1 2 ≤ s ≤ t ≤ 1 . We conjecture that this inequality is also true for all unitarily invariant norms, and we affirmatively settle this conjecture for the case s = 1 2 and t = 1 .