A norm inequality on noncommutative symmetric spaces related to a question of Bourin

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.