Recently, T. Furuta and M. Giga{\cite{FG}} gave the complementary
result of Kantorovich type order preserving inequalities by
Mi\'{c}i\'{c}-Pe\v{c}ari\'{c}-Seo{\cite{S2}}. In this note, we shall
show a difference version of T. Furuta and M. Giga's result as
follows: Let $A$ and $B$ be positive operators on a Hilbert space
$H$ such that $A \ge B \ge 0$ and $MI \ge A \ge m I>0 $ for some
scalars $M>m>0$. If $p>1$ and $q>1$, then the following inequality
holds:
$$ B^p \le A^q + C(m ,M, p, q)I,$$ where $C(m, M, p, q) =
\left\{\frac{M^p - m^p }{q(M -m)}\right\}^{\frac{q}{q-1}} (q-1) +
\frac{m ^p M - m M ^p }{M-m}$ for $m \le \left\{\frac{M^p -
m^p}{q(M-m)}\right\}^\frac{1}{q-1} \le M$. In addition, we obtain
Kantorovich type inequalities for the chaotic order.