Uchiyama discussed the Furuta inequality from 
the viewpoint of the Jensen inequality.  
Recently Furuta and Kamei improved it as follows:
Suppose that $A,B,C > 0$ and $r, s \ge 0$. 
If $A^t \ll B^t~\nabla_\mu~C^t$ for all $t \ge 0$, 
then $$ f(t) =  A^{-r}~\sharp_{\frac{s+r}{t+r}}~
(B^t ~\nabla_\mu~C^t)$$ is an increasing function 
of $t \ge s$.
On the other hand, if $A^t \ll B^t~!_\mu~C^t$ 
for all $t \ge 0$, then $$ h(t) =  A^{-r}~
\sharp_{\frac{s+r}{t+r}}~(B^t ~!_\mu~C^t)$$
is a decreasing function of $t \ge s$.

In this note, we pay our attention to the assumptions 
in above and point out that the operator function 
$  F(s)=((1-\mu)A^s + \mu B^s)^{\frac{1}{s}} ~
(s \in \Bbb R)$ for given $A, B > 0$ and $\mu \in [0,1]$ 
is monotone increasing under the chaotic order $X \gg Y$ 
defined by $\log X \ge \log Y$ and consequently 
s-$\lim_{h \rightarrow 0} F(h) = e^{(1-\mu)\log B + 
\mu \log C}$.This means that we can see another geometric 
mean $ B~\dia_\mu~ C = e^{(1-\mu)\log B + \mu \log C}  $ 
in the Furuta inequality.  Moreover we consider 
Uchiyama's result in a general setting.