We show the Furuta inequality and grand one are useful to interpolate Kantorovich

type operator inequalities under usual order and chaotic order. If $M \ge A 

\ge m >0$ and $A^t \ge B^t$ for $t \in [0,\ 1]$, then 

$$\frac{(M^{p-t}+m^{p-t})^2}{4M^{p-t}m^{p-t}}A^p \ge B^p\ \ \ \ for\ \ all\ \ p 

\ge 2t. $$

The case where $t=1$ corresponds to the usual operator order and the case $t=0$ 

to the chaotic order.