Recently, Furuta has shown several inequalities with forms of 
grand Furuta inequality and clarified difference between 
chaotic order and usual order of positive operators. 
Here the chaotic order $A \gg B$ for positive invertible 
operators $A$ and $B$ is defined by $\log A \ge \log B$.  
In this note, we present the following inequalities which are 
based on the Furuta inequality for chaotic order and can be 
regarded as interpolations of recent Furuta's results. 
Among others, we obtain the following:\\
 Let $A,\ B$ be positive invertible operators satisfying   $A \gg B$.\\
{\rm (1)} If $0 \le \delta \le \beta \le p$ and $r \ge 0$, then
$$B^{\delta} \ge A^{-r}\ \sharp_{\frac{\delta+r}{\beta+r}}\ B^{\beta} 
\ge A^{-r}\ \sharp_{\frac{\delta+r}{\beta+r}}\ (A^{-r}\ 
\sharp_{\frac{\beta+r}{p+r}}\ B^p).$$
{\rm (2)} If $|\delta| \le p \le \beta \le 2p$ and $r \ge t \ge |\delta|$ 
for some $\delta \in \mathbb{R}$, then\\
$$A^{-t}\ \sharp_{\frac{\delta+t}{p+t}}\ B^p \ge A^{-r}\ 
\sharp_{\frac{\delta+r}{\beta+r}}\ (A^{-t}\ \natural_{\frac{\beta+t}{p+t}}\ 
B^p) \ge A^{-r}\ \sharp_{\frac{\delta+r}{\beta+r}}\ B^{\beta}.$$