A Unitarily Invariant Norm Inequality for Positive Semidefinite Matrices and a Question of Bourin

In this article, we obtain a new unitarily invariant norm inequality for positive semidefinite matrices. In fact, we prove that if A and B are positive semidefinite matrices and t is an element of[3/4, 1], then vertical bar vertical bar vertical bar B1-t A(2t-1) B1-t + A(1-t) B2t-1 A(1-t)vertical bar vertical bar vertical bar <= 2(4(t-3/4)) vertical bar vertical bar vertical bar A + B vertical bar vertical bar vertical bar. The significance of this result is that it is sharper than an earlier norm inequality and closely related to an open question of Bourin. In particular, this inequality gives a way to settle Bourin's question for t = 1/4 and 3/4, which is a result due to Hayajneh and Kittaneh [9].