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Let A and B be positive semidefinite matrices. It is shown that ∥∥AsBp +BqAt∥∥2 ∥∥AsBp +AtBq∥∥2 for all positive real numbers s,t, p,q for which ∣∣∣ s s+ t − 1 2 ∣∣∣+ ∣∣∣ p p+q − 1 2 ∣∣∣ 1 2 . This is a generalization of a recent inequality proved by Bhatia for the special case s = q , t = p with

Remarks on some norm inequalities for positive semidefinite matrices and questions of Bourin