On some inequalities for positive operators

Abstract

In the present paper, our aim is to refine some operator inequalities for arbitrary operator means and positive maps. For example, If $$A,B\in {{\mathbb {B}}}({{\mathscr {H}}})$$ be two invertible positive operators with this condition that $$0< m^{'}\le B \le m < M\le A \le M^{'}$$ for some positive real numbers $$m,m^{'},M,M^{'},$$ $$\sigma _{1}$$ and $$\sigma _{2}$$ be two arbitrary means between geometric and arithmetic means and $$\Phi$$ be a positive linear map. Then $$\begin{aligned} \Phi ^{2}(A \sigma _{1} B)\le \left( \frac{ K(h^{'}) }{ \Theta _{\frac{1}{2}}(h)} \right) ^{2}\Phi ^{2}(A \sigma _{2} B), \end{aligned}$$ where $$K(h^{'})=\frac{(h^{'}+1)^{2}}{4h^{'}}$$ with $$h^{'}=\frac{m^{'}}{M^{'}}$$ is the Kantorovich constant and $$\Theta _{\frac{1}{2}}(h)=1+\frac{\sqrt{2}(h-1)^{2}}{4(h+1)^{\frac{3}{2}}}$$ with $$h=\frac{m}{M}$$ is decreasing.