In the discussion of the equivalence between Furuta inequality and Ando-Hiai inequality, the key is the inequality that if 
$A \ge B > 0$, then 
$$
A^{-r}\ \sharp_{\frac{ r}{p+r }}\ B^p \le I \quad {\rm for} \quad p, r \ge 1.  
$$
Here $\sharp_\alpha$ is the $\alpha$-geometric mean in the sense of Kubo-Ando.
In this note, we assume that 
$A^{-r}\ \sharp_{\frac{ r}{p+r }}\ B^p \le I$ for some $p, r \ge 1$ instead of $A \ge B > 0$.  Then we show that
$$ A^{-r}\ \sharp_{\frac{\delta+r}{p+r}}\ B^p \le A^{-r}\ \sharp_{\frac{\delta+r}{\mu+r}}\ B^{\mu}
 %\leqno{\rm (i)}
 \quad for \ \ 0 \le \delta \le \mu \le p,$$ and for each $t \in [0,r]$
$$A^{-r}\ \sharp_{\frac{\delta+r}{p+r}}\ B^p \le A^{-t}\ \sharp_{\frac{\delta+t}{p+t}}\ B^p
 \quad for \ -t \le \delta \le p.$$
% \left\{
%\begin{array}{rl}
%A^{\delta} & \mbox{$-t \le \delta \le 0$}\\
%B^{\delta} & \mbox{$0 \le \delta \le p$.}
%\end{array}\right.
%   \leqno{\rm (ii)}$$
As an application, we discuss recent development of grand Furuta inequality due to Furuta himself.