Uchiyama gave a generalization of the grand Furuta inequality and 
Furuta discussed it based on his previous result.  Motivated by 
such discussions, we consider grand Furuta type operator inequalities 
of 3 variables, whose hidden key is the chaotic order, i.e., 
$\log A \ge \log B$ for positive invertible operators $A$ and $B$.  
Among others, Uchiyama's theorem and Furuta's theorem are appeared 
as follows:  For $A \ge B \ge C > 0$ and $0 \le t \le 1 \le p$
%(1)
$$  B \ge C \ge (B^t\ \natural_{\frac{\beta-t}{p-t}}\ C^p)^{\frac{1}{\beta}} 
\ge B^{\frac{t}{2}}A^{-t}B^{\frac{t}{2}}\ \sharp_{\frac{1}{\beta}}\ (B^t\ 
\natural_{\frac{\beta-t}{p-t}}\ C^p) \ge B^{\frac{t}{2}}A^{-r}B^{\frac{t}{2}}\ 
\sharp_{\frac{1-t+r}{\beta-t+r}}\ (B^t\ \natural_{\frac{\beta-t}{p-t}}\ C^p) $$
and
%(2)
$$ B \ge C \ge B^{\frac{t}{2}}A^{-t}B^{\frac{t}{2}}\ \sharp_{\frac{1}{p}}\ C^p \ge B^{\frac{t}{2}}A^{-r}B^{\frac{t}{2}}\ \sharp_{\frac{1-t+r}{p-t+r}}\ C^p \ge 
B^{\frac{t}{2}}A^{-r}B^{\frac{t}{2}}\ \sharp_{\frac{1-t+r}{\beta-t+r}}\ (B^{t}\
 \natural_{\frac{\beta-t}{p-t}}\ C^p) $$ 
hold for $\beta \ge p$ and $r \ge t$.\\
As a complement to our preceding inequality, we give an inequality for 
$A \gg B \gg C$ and $t \ge 0,\ 0 \le p \le \beta \le 2p$,
$$C^{-t}\ \sharp_{\frac{p+t}{\beta+t}}\ B^{\beta} \ge B^p \ge (C^{-t}\ 
\natural_{\frac{\beta+t}{p+t}}\ B^p)^{\frac{p}{\beta}} \ge A^{-t}\ 
\sharp_{\frac{p+t}{\beta+t}}\ (C^{-t}\ \natural_{\frac{\beta+t}{p+t}}\ B^p).$$