New Inequalities of the Kantorovich Type With Two Negative Parameters

We show the following result: Let $A,B\in $\mathbb{B}$\left( $\mathcal{H}$ \right)$betwostrictlypositiveoperatorssuchthat$A\le B$and$m{{$\mathbf{1}$}{$\mathcal{H}$}}\le B\le M{{$\mathbf{1}$}{$\mathcal{H}$}}$ for some scalars $0<m<M$. Then $$B^p\le \exp (\frac{M{\mathbf{1}}_{\mathcal{H}}-B}{M-m}\ln m^p+\frac{B-m{\mathbf{1}}_{\mathcal{H}}}{M-m}\ln M^p)\le K(m,M,p,q)A^q\text{ for }p\le 0,-1\le q\le 0$$ where $K(m,M,p,q)$ is the generalized Kantorovich constant with two parameters. In addition, we obtain Kantorovich type inequalities for the chaotic order.