Improvements of some operator inequalities involving positive linear maps via the Kantorovich constant

Abstract

We present some operator inequalities for positive linear maps that generalize and improve the derived results in some recent years. For instant, if $A$ and $B$ are positive operators and $m,m^{{}^{\prime }},M,M^{{}^{\prime }}$ are positive real numbers satisfying either one of the condition $0<m\le B\le m^{{}^{\prime }}<M^{{}^{\prime }}\le A\le M$ or $0<m\le A\le m^{{}^{\prime }}<M^{{}^{\prime }}\le B\le M$, then \begin{align*} Φ^{p} \big(A \nabla _{v} B+2 r Mm (A^{-1}\nabla B^{-1}- &A^{-1} \sharp B^{-1} )\big)\\ & \leq \left( \frac{K(h)}{ 4^{\frac{2}{p}-1} K^{r_{1}} \left( \sqrt {h^{'}}\right)} \right) ^{p} Φ^{p} (A \sharp_ν B) \end{align*} and \begin{align*} Φ^{p} \big(A \nabla _{v} B+2 r Mm (A^{-1}\nabla B^{-1}-& A^{-1} \sharp B^{-1} )\big) \\ &\leq \left( \frac{K(h)}{ 4^{\frac{2}{p}-1} K^{r_{1}}\left( \sqrt {h^{'}}\right)}\right) ^{p} (Φ(A) \sharp_ν Φ(B))^{p}, \end{align*} where $\Phi$ is a positive unital linear map, $0\le \nu \le 1$, $p\ge 2,$ $r=\min \{\nu ,1-\nu \},$ $h=\frac{M}{m},$ $h^{{}^{\prime }}=\frac{M^{{}^{\prime }}}{m^{{}^{\prime }}}$, $K(h)=\frac{(1+h{)}^2}{4h}$ and $r_1=\min \{2r,1-2r\}.$ We also obtain a reverse of the Ando inequality for positive linear maps via the Kantorovich constant.

To appear in Houston journal of mathematics