A Lieb-Thirring inequality for singular values

Let $A$ and $B$ be positive semidefinite matrices. We investigate the conditions under which the Lieb-Thirring inequality can be extended to singular values. That is, for which values of $p$ does the majorisation $σ(B^p A^p) \prec_w σ((BA)^p)$ hold, and for which values its reversed inequality $σ(B^p A^p) \succ_w σ((BA)^p)$.

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