Complete form of Furuta inequality

Let $A$ and $B$ be bounded linear operators on a Hilbert space satisfying $A \ge B \ge 0$. The well-known Furuta inequality is given as follows: Let $r\ge 0$ and $p> 0$; then $A^{\frac{r}{2}} A^{\min\{1,p\}} A^{\frac{r}{2}} \ge(A^{\frac{r}{2}} B^p A^{\frac{r}{2}})^{\frac{\min\{1,p\}+r}{p+r}}$. In order to give a self-contained proof of it, Furuta (1989) proved that if $1\geq r\ge 0$, $p>p_{0}> 0$ and $2p_{0}+r\geq p>p_{0}$, then $(A^{\frac{r}{2}} B^{p_{0}} A^{\frac{r}{2}})^{\frac{p+r}{p_{0}+r}} \ge(A^{\frac{r}{2}} B^p A^{\frac{r}{2}})^{\frac{p+r}{p+r}}$.

This paper aims to show a sharpening of Furuta (1989): Let $r\ge 0$, $p_{0}> 0$ and $s=\min\{p, 2p_{0}+\min\{1,r\}\}$; then $(A^{\frac{r}{2}} B^{p_{0}} A^{\frac{r}{2}})^{\frac{s+r}{p_{0}+r}} \ge(A^{\frac{r}{2}} B^p A^{\frac{r}{2}})^{\frac{s+r}{p+r}}$. We call it the complete form of Furuta inequality because the case $p_{0}=1$ of it implies the essential part ($p>1$) of Furuta inequality for $\frac{1+r}{s+r}\in (0,1]$ by the famous Löwner-Heinz inequality. Afterwards, the optimality of the outer exponent of the complete form is considered. Lastly, we give some applications of the complete form to Aluthge transformation.