Bourin-type inequalities for $τ$-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 $τ$ be a fixed faithful normal semifinite trace on $\mathcal{M}$.Let $E_τ$ 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 $a,b\in E_τ$ and $t\in[0,1]$, $$ |a^t b^{1-t}+b^t a^{1-t}|{E_τ}\le 2^{\max{2|t-1/2|-1/2,;0}};|a+b|{E_τ}. $$ In particular, we obtain the sharp constant $1$ for $t\in[1/4,3/4]$: $$ |a^t b^{1-t}+b^t a^{1-t}|{E_τ}\le |a+b|{E_τ}. $$ 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

💡 Research Summary

The paper investigates Bourin‑type inequalities for τ‑measurable operators within fully symmetric spaces associated with a semifinite von Neumann algebra 𝔐⊂B(ℋ) equipped with a faithful normal semifinite trace τ. Let E be a fully symmetric Banach function space on