Inspired by a recent Furuta's result, we give a
generalization of parametrized grand Furuta inequality; if $A \ge B >
0$, then
$$A^u\ \sharp_{\frac{\delta-u}{\beta-u}}\ (A^t
 \natural_{\frac{\beta-t}{p-t}}\ B^{p}) \le (A^{t}
 \natural_{\frac{\beta-t}{p-t}}\ B^{p})^{\frac{\delta}{\beta}}$$
for $t \in [0,1], 0 \le t < p \le \beta,\ u \le 0\ and\ \delta
\in[0,\beta].$
As an application we discuss the monotonicity of an opertor function for
$A \ge B$ by
$$H_{p,\delta,t}(A,B,u,\beta)=A^{u}\ \sharp_{\frac{\delta-u}{\beta-u}}
 (A^{t}\ \natural_{\frac{\beta-t}{p-t}}\ B^{p});$$
it is increasing for $u \le 0$ and decreasing for $\beta \ge p$ where $0
\le t < p \le \beta$, $u \le 0$ and $\delta \in [0,\beta]$.\\