A note on Audenaert interpolation inequality

Let A, B, X. M(n) and |||.||| be a unitarily invariant norm. Weintroduce a log-convex (and hence a convex) function g on the interval [ 0, 1] such that g is decreasing on [ 0, 1/2], increasing on [ 1/2, 1] and attains its minimum at 1/2. Moreover, ||| AXB *||| 2 = g(1/2) = g(t) = ||| tA * AX + (1 -t) XB * B||| |||(1 -t) A * AX + tXB * B|||, for t. [ 0, 1]. Related interpolating inequalities are also proved. This implies an improvement of a recent result of Audenaert.