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 $[0,\mathrm{\infty })$.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 $t\in [0,1]$, $$\mathrm{\parallel }{a}^t{b}^{1-t}+b^t{a}^{1-t}{\mathrm{\parallel }}_{E_{\tau }}\le 2^{\max \{2\mathrm{\mid }t-1\mathrm{/}2\mathrm{\mid }-1\mathrm{/}2,0\}}\mathrm{\parallel }a+b{\mathrm{\parallel }}_{E_{\tau }}\mathrm{.}$$ In particular, we obtain the sharp constant $1$ for $t\in [1\mathrm{/}4,3\mathrm{/}4]$: $$\mathrm{\parallel }{a}^t{b}^{1-t}+b^t{a}^{1-t}{\mathrm{\parallel }}_{E_{\tau }}\le \mathrm{\parallel }a+b{\mathrm{\parallel }}_{E_{\tau }}\mathrm{.}$$ This extends the work of Kittaneh–Ricard in \emph{Linear Algebra Appl.} $\mathbf{\text{710}}$ (2025), 356–362 and covers the results of Liu–He–Zhao in \emph{Acta Math. Sci. Ser. B (Engl. Ed.)} $\mathbf{\text{46}}$ (2026), 62–68