Improvements of some operator inequalities involving positive linear maps via the Kantorovich constant (1801.02030v1)

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 $m,m^{'},M,M^{'}$ are positive real numbers satisfying either one of the condition $ 0<m \leq B \leq m^{'} <M^{'} \leq A \leq M $ or $0<m \leq A \leq m^{'} <M^{'} \leq B \leq M$, then \begin{align*} \Phi ^{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} \Phi^{p} (A \sharp_{\nu} B) \end{align*} and \begin{align*} \Phi ^{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} (\Phi(A) \sharp_{\nu} \Phi (B))^{p}, \end{align*} where $\Phi$ is a positive unital linear map, $ 0 \leq \nu \leq 1$, $p \geq 2,$ $r=\min{\nu,1-\nu},$ $h=\frac{M}{m},$ $h^{'}=\frac{M^{'}}{m^{'}}$, $K(h)=\frac{(1+h)^{2}}{4h}$ and $r_{1}=\min{2r,1-2r}.$ We also obtain a reverse of the Ando inequality for positive linear maps via the Kantorovich constant.