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MR897075 47A60 (15A45 47B15)
Furuta, Takayuki $A\geq B\geq 0$ assures $(B^rA^pB^r)^{1/q}\geq B^{(p+2r)/q}$ for $r\geq 0$, $p\geq 0$, $q\geq 1$ with $(1+2r)q\geq p+2r$. Proc. Amer. Math. Soc. 101 (1987), no. 1, 85–88.
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