SOME EXTENSIONS OF KANTOROVICH TYPE INEQUALITIES (Recent Topics on Operator inequalities)

We consider Kantorovich type inequalities for bounded strictly positive operators on a Hilbelt space.$\mathrm{M}\mathrm{i}\acute{\mathrm{c}}\mathrm{i}\acute{\mathrm{c}}-\mathrm{P}\mathrm{e}\check{\mathrm{c}}\mathrm{a}\mathrm{r}\mathrm{i}\acute{\mathrm{c}}$ -Seo recently obtained Kantorovich type inequalities between $A^{q}$ and $B^{p}$ for the case $p>1$ , $q>$ $1$ under the assumption $A\geq B.$ We extend it to more generalized Kan- torovich type inequalities between $(Tx, x)^{q}$ and $(T^{\mathrm{p}}x, x)$ for the case (a) $p>1,q>1$ , (b) $p<0$ , $q<0$ , (c) $0<p<1,0<q<1.$We further prove that these results are applied to the case chaotic order.