Very recently, Furuta obtained the grand Furuta inequality which is a parameteric formula interpolating the Furuta inequality and the Ando-Hiai inequality as follows : If
A
≥
B
≥
0
A \ge B \ge 0
and
A
A
is invertible, then for each
t
∈
[
0
,
1
]
t \in [0,1]
,
F
p
,
t
(
A
,
B
,
r
,
s
)
=
A
−
r
/
2
{
A
r
/
2
(
A
−
t
/
2
B
p
A
−
t
/
2
)
s
A
r
/
2
}
1
−
t
+
r
(
p
−
t
)
s
+
r
A
−
r
/
2
\begin{equation*}F_{p,t}(A,B,r,s) = A^{-r/2}\{A^{r/2}(A^{-t/2}B^{p}A^{-t/2})^{s}A ^{r/2}\}^{\frac {1-t+r}{(p-t)s+r}}A^{-r/2} \end{equation*}
is a decreasing function of both
r
r
and
s
s
for all
r
≥
t
,
p
≥
1
r \ge t, ~p \ge 1
and
s
≥
1
s \ge 1
. In this note, we employ a mean theoretic approach to the grand Furuta inequality. Consequently we propose a basic inequality, by which we present a simple proof of the grand Furuta inequality.