根据Lin的思想，证明了：若$A$和$B$是两个正算子，使得$0<mI\le A\le m'I\le M'I\le B\le MI$，则\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*}和\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*}，其中$K\left( h \right)=\frac{{{\left( h+1 \right)}^{2}}}{4h}$和$h=\frac{M}{m}$和$Φ$是一个正单位线性映射.
英文摘要：
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*} {{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 $Φ$ is a positive unital linear map.
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