Simplified proof of an order preserving operator inequality (Q1281902)

Let \(A\) and \(B\) be bounded linear operators on a Hilbert space. The author gives a simplified proof of the theorem: If \(A\geq B\geq 0\), with \(A>0\), then for \(1\geq q \geq t\geq 0\) and \(p\geq q\) the inequality \(A^{q-t+r}\geq \{A^{r/2}(A^{-t/2}B^pA^{-t/2})^sA^{r/2} \}^{(q-t+r)/[(p-t)s+r]}\) holds for \(s\geq 1\) and \(r\geq t\). As the author mentioned this theorem was proved in the earlier preprint: \textit{T.Furuta, T.Yamazaki, M.Yanagida}: Order preserving operator inequalities via Furuta inequality, but he gives no information where this preprint is published.