$A \ge B \ge 0$ ensures $(A^{\frac{r}{2}}A^pA^{\frac{r}{2}})^{\frac{1}{q}} \ge (A^{\frac{r}{2}}B^pA^{\frac{r}{2}})^{\frac{1}{q}}$ for $p \ge 0,q \ge 1,r \ge 0$ with $(1+r)q \ge p+r$ and brief survey of its recent applications (Inequalities on Linear Opera
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