Complementary Inequalities to Improved AM-GM Inequality

Abstract

Following an idea of Lin, we prove that if $A$ and $B$ be two positive operators such that $0<mI\le A\le m^{\prime }I\le M^{\prime }I\le B\le MI$, then \begin{equation*} {{Φ}^{2}}\left( \frac{A+B}{2} \right)\le \frac{{{K}^{2}}\left( h \right)}{{{\left( 1+\frac{{{\left( \log \frac{M'}{m'} \right)}^{2}}}{8} \right)}^{2}}}{{Φ}^{2}}\left( A\#B \right), \end{equation*} and \begin{equation*} {{Φ}^{2}}\left( \frac{A+B}{2} \right)\le \frac{{{K}^{2}}\left( h \right)}{{{\left( 1+\frac{{{\left( \log \frac{M'}{m'} \right)}^{2}}}{8} \right)}^{2}}}{{\left( Φ\left( A \right)\#Φ\left( B \right) \right)}^{2}}, \end{equation*} where $K\left(h\right)=\frac{{\left(h+1\right)}^2}{4h}$ and $h=\frac{M}{m}$ and $\Phi$ is a positive unital linear map.

to appear in Acta Math. Sin