The grand Furuta inequality says that 
if $A \ge B > 0$, then 
\begin{equation}
 A^{1+r-t } \geq 
\{ A^{\frac{r}{2}}(A^{-\frac{t}{2}}B^pA^{-\frac{t}{2}})^s
A^{\frac{r}{2}} \} ^{\frac{1+r-t }{(p-t)s+r}} 
 \tag{G}  \end{equation}
holds for all  $p\geq 1$, $r \geq t$,  $s\geq 1$ and  
$t\in [0,1]$.
Very recently Uchiyama gave an extension of 
the grand Furuta inequality as follows:
If $A \ge B \ge C > 0$, then 
\begin{equation}
 A^{1+r-t } \geq \{ A^{\frac{r}{2}}
(B^{-\frac{t}{2}}C^pB^{-\frac{t}{2}})^s
A^{\frac{r}{2}} \} ^{\frac{1+r-t }{(p-t)s+r}} 
 \tag{U}  \end{equation}
holds for all  $p\geq 1$, $r \geq t$,  $s\geq 1$ and  
$t\in [0,1]$.  
The purpose of this short note is to propose 
a simplifed proof of Uchiyama's extension. 
 Moreover we pose a variant of the grand Furuta 
inequality motivated by Uchiyama's idea.