  Following an idea of Lin, we prove that if $A$ and $B$ be two positive
operators such that $0<mI\le A\le m'I\le M'I\le B\le MI$, 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.
