On a simple range of parameters of the grand Furuta inequality (Q2855678)

The grand Furuta inequality, which interpolates the Furuta inequality and an inequality equivalent to a log-majorization theorem by \textit{T. Ando} and \textit{F. Hiai} [Linear Algebra Appl. 197--198, 113--131 (1994; Zbl 0793.15011)], states that, if \(A \geq B \geq 0\) and \(A\) is invertible, then, for each \(t \in [0,1]\), \(p \geq 1\), \(s \geq 1\) and \(r \geq t\), it holds that NEWLINE\[NEWLINEA^{1-t+r}\geq (A^{r/2}(A^{-t/2}B^pA^{-t/2})^sA^{r/2})^{(1-t+r)/((p-t)s+r)}.NEWLINE\]NEWLINE \textit{K. Tanahashi} [Proc. Am. Math. Soc. 128, No. 2, 511--519 (2000; Zbl 0943.47016)] showed that the outside powers in this theorem are the best possible. \textit{T. Yamazaki} [Math. Inequal. Appl. 2, No. 3, 473--477 (1999; Zbl 0938.47015)] presented a simplified proof for the claim of Tanahashi. Using some arguments from [\textit{T. Yamazaki}, Math. Inequal. Appl. 2, No.3, 473--477 (1999; Zbl 0938.47015)], the author shows that, if \(t \in (0,1)\), \(p \geq 1\), \(0<s < \frac{1-t}{p-t}\) and \(r >0\), then there exist operators \(A, B\) with \(0<B\leq A\) which do not satisfy the grand Furuta inequality; see also \textit{T. Koizumi} and \textit{K. Watanabe} [Cent. Eur. J. Math. 11, No. 2, 368--375 (2013; Zbl 1269.47016)].