On Araki-Type Trace Inequalities

In this paper, we prove a trace inequality $\text{Tr}[f(A)A^s{B}^s]\le \text{Tr}[f(A)(A^{1\mathrm{/}2}B{A}^{1\mathrm{/}2}{)}^s]$ for any positive and monotone increasing function $f$, $s\in [0,1]$, and positive semi-definite matrices $A$ and $B$. On the other hand, for $s\in [0,1]$ such that the map $x\mapsto x^{s}g(x)$ is positive and decreasing, then $\text{Tr}[g(A)(A^{1\mathrm{/}2}B{A}^{1\mathrm{/}2}{)}^s]\le \text{Tr}[g(A)A^s{B}^s]$.