The best possibility of the grand Furuta inequality

Let $A, B \in B(H)$ be invertible bounded linear operators on a Hilbert space $H$ satisfying $O\leq B \leq A$ , and let $p, r, s , t$ be real numbers satisfying $1 < s, 0 < t < 1 , t \leq r , 1 \leq p .$ Furuta showed that if $0 < \alpha \leq \dfrac{ 1-t+r}{ (p-t)s + r}$, then $\left\{ A^{\frac{r}{2}} \left( A^{ -\frac{t}{2}} B^{p} A^{ -\frac{t}{2}} \right)^{s} A^{\frac{r}{2}} \right\}^{\alpha } \leq A^{ \left\{ (p-t)s + r \right\} \alpha }$. This inequality is called the grand Furuta inequality, which interpolates the Furuta inequality $(t=0)$
and the Ando-Hiai inequality ( $t=1, r = s$ ).

In this paper, we show the grand Furuta inequality is best possible in the following sense: that is, if $\dfrac{ 1-t+r}{ (p-t)s + r} < \alpha$, then there exist invertible matrices $A,B$ with $O\leq B \leq A$ which do not satisfy $\left\{ A^{\frac{r}{2}} \left( A^{ -\frac{t}{2}} B^{p} A^{ -\frac{t}{2}} \right)^{s} A^{\frac{r}{2}} \right\}^{\alpha } \leq A^{ \left\{ (p-t)s + r \right\} \alpha }$.