  In this note, we study a question introduced by Bourin \cite{2009Matrix} and
partially solve the question of Bourin. In fact, for
t\in[0,\frac{1}{4}]\cup[\frac{3}{4},1], we show that
|||x^{t}y^{1-t}+y^{t}x^{1-t}|||\leq|||x+y|||, where
x,y\in\mathbb{M}_{n}(\mathbb{C})^+ and \||\cdot\|| is the unitarily invariant
norm. Moreover, we prove that the above inequality holds on noncommutative
fully symmetric spaces.
