Bourin-type inequalities for $\tau$-measurable operators in fully symmetric spaces

Let $\mathcal{M}\subset B(\mathcal{H})$ be a semifinite von Neumann algebra, where $B(\mathcal{H})$ denotes the algebra of all bounded linear operators on a Hilbert space $\mathcal{H}$, and let $\tau$ be a fixed faithful normal semifinite trace on $\mathcal{M}$.Let $E_\tau$ be the fully symmetric space associated with a fully symmetric Banach function space $E$ on $[0,\infty)$.Using a complex interpolation argument based on the three-lines theorem on a strip, we show that for positive operators $a,b\in E_\tau$ and $t\in[0,1]$, $$\|a^t b^{1-t}+b^t a^{1-t}\|_{E_\tau}\le 2^{\max\{2|t-1/2|-1/2,\;0\}}\;\|a+b\|_{E_\tau}.$$ In particular, we obtain the sharp constant $1$ for $t\in[1/4,3/4]$: $$\|a^t b^{1-t}+b^t a^{1-t}\|_{E_\tau}\le \|a+b\|_{E_\tau}.$$ This extends the work of Kittaneh--Ricard in \emph{Linear Algebra Appl.} \textbf{710} (2025), 356--362 and covers the results of Liu--He--Zhao in \emph{Acta Math. Sci. Ser. B (Engl. Ed.)} \textbf{46} (2026), 62--68