In our previous note, we showed the difference version of
Kantorovich type inequality with two positive parameters under the
usual order. As a continuation of our preceding note, we shall show
the difference type Kantorovich inequalities with two positive
parameters under the chaotic order in terms of a two parameters version 
of the Mond-Shisha difference: Let $A$ and $B$ be positive and
invertible operators on a Hilbert space $H$ such that $MI \ge B \ge
mI$ for some scalars $0<m<M$ and $h=\frac{M}{m}.$  Then the
following assertions are mutually equivalent:

\begin{list}{}{\leftmargin=30pt\labelwidth=20pt\labelsep=10pt\parsep=0pt}
\item[\rm(1)] $\log A \ge \log B,$
\item[\rm(2)] $A^q +\frac {(M^{p+q} -m^{p+q})^2
-4m^qM^q(M^p-m^p)(M^q-m^q)}{4m^q(M^q-m^q)^2} I \ge B^p$ \quad
\mbox{for all } $p,\ q>0,$
\item[\rm(3)] $A^q + {\frac{p}{q}}{\frac {M^p - m^p}{\log M^p -\log m^p}}
\log\left(m^{p-q} {\frac{(h^p-1)h^{\frac{q}{h^p-1}}}{{e}{q} \log
h}}\right)I \ge B^p$ \quad \mbox{for all } $p,\ q>0$.
\end{list}