$A \ge B \ge 0$ ensures $(B^r A^p B^r)^{1/q} \ge (B^r B^p B^r)^{1/q}$ for $r \ge 0, p \ge 0, q \ge 1$ with (1 + 2r)q $\ge$ p + 2r and its applications(Linear Operators and Inequalities)
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