As an application of both the Furuta inequality and the grand Furuta inequality, 

we shall show Kantorovich type order preserving operator inequalities under the 

usual order by means of the Ky Fan-Furuta constant, which are parametrized some 

results on the chaotic order obtained in recent papers by Furuta-Seo and 

Hashimoto-Yamazaki. Among others, we show that for each $\delta \in [0,1]$

$$A^{\delta }\geq B^{\delta }\quad \text{if and only if} 

\quad K_+(m^r, M^r, 1+\frac{p-\delta }{r})A^p\geq B^p 

\quad \text{for $p\geq \delta $ and $r\geq \delta $,}

$$ where the case $\delta =0$ means the chaotic order.