Complementary Inequalities to Improved AM-GM Inequality

Following an idea of Lin, we prove that if $A$ and $B$ be two positive operators such that $0<mI\le A\le m^{\mathrm{\prime }}I\le M^{\mathrm{\prime }}I\le B\le MI$, then \begin{equation*} {{\Phi }^{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}}}{{\Phi }^{2}}\left( A#B \right), \end{equation*} and \begin{equation*} {{\Phi }^{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( \Phi \left( A \right)#\Phi \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.