Abstract. This paper is to discuss the Furuta inequality and the Furuta-type operator function $f_{r,s}(p)=(A^{{r}/{2}}B^pA^{{r}/{2}})^{(s+r)/(p+r)}$ under $\log A\geq \log B$. Firstly, we provide a direct proof of the best possibility of the Furuta inequality with negative powers by using Tanahashi's two invertible operators. Secondly, it is well known that $f_{r,s}(p)$ is decreasing for $p\geq \max \{s,0\} $ when $r>0$ and $s>-r$. We show that, for each $r>0$ and $s>-r$, the monotone interval $[\max \{s,0\},\infty )$ is the unique one for the function $f_{r,s}(p)$ under chaotic order in the interval $[-r,\infty )$.

AMS Subject Classification (1991): 47A63, 47B15

Keyword(s): positive operator, chaotic order, Furuta inequality

Received September 14, 2006, and in revised form January 8, 2007. (Registered under 6456/2009.)