Let $B, R\in M_{n}(\mathbb{C})$ hermitian and $B$ positive semidefinite. Let $s \in \mathbb{R}$ and $s \ge 0$ .
Does then hold $Tr[B^s (I + B R^2 B)^{-1}] \le Tr[B^s (I + R B^2 R)^{-1}]$ ?
See also A conjectured trace inequality for some products of powers of matrices .
