In this note, we obtain more precise estimations than the constants
are given in the paper by M.Fujii, E.Kamei and Y.Seo, {\it
Kantorovich type operator inequalities via grand Furuta inequality},
Sci. Math., {\bf 3} (2000), 263--272. Among other, we show that the
following statements are mutually equivalent for each $\delta \in
(0,1]$:

\noindent (i) \quad \qquad  \qquad \quad $
K(m^{\frac{(p-\delta)s}{n}},M^{\frac{(p-\delta)s}{n}},n+1)^{\frac{1}{s}}
A^{p} \geq B^{p}$

\noindent \qquad \qquad \qquad for any $n>0$, $s \geq 1$, $p \geq
\delta$ with $ (p-\delta )s \geq n \delta $.


\noindent (ii) \quad \qquad  \qquad \quad
$K(m^{\delta},M^{\delta},\frac{p}{\delta}) A^{p} \geq B^{p} \qquad
\mbox{for any $p \geq \delta$ }$.

\noindent For each $\delta \in (0,1]$
$$K(m^{\frac{(p-\delta)s}{n}},M^{\frac{(p-\delta)s}{n}},n+1)^{\frac{1}{s}}
\geq K(m^{\delta},M^{\delta},\frac{p}{\delta})$$ holds for any
$n>0$, $s \geq 1$, $p \geq \delta$ such that $ (p-\delta )s \geq n
\delta $.