In \cite{K}, one of the authors showed Kantorovich type inequalities
with two positive parameters, which is a difference version of
Furuta-Giga's one \cite{Fg}. In this note, we show the following
order for $p\leqq-1$: Let $A\ge B>0$ and $MI \ge A \ge m I>0 $ for some
positive numbers $M>m>0$. Then
$$B^p+ C(m,M,p)I \ge B^p +C\left ({\frac{1}{M}},{\frac{1}{m}}, -p \right)I\ge A^p$$
holds for $p\leqq-1$ where the constant is defined as
$$C(m,M,p)\equiv(p-1)\left({\frac{M^p-m^p}{p(M-m)}}\right)^{\frac{p}{p-1}}+{\frac{Mm^p-M^pm}{M-m}}.$$
We also show a similar inequality for the chaotic order: Let $A$ and
$B$ positive operators on a Hilbert space with $MI\ge A\ge mI>0$ for
some positive numbers $M>m>0$. If $\log A\ge\log B$, then
$$B^p+{\frac{M}{m}}(m^p-M^p)I\ge B^p+C\(\frac{1}{M},{\frac{1}{m}},1-p\)MI\ge A^p$$
for $p\leqq0$.