Complementary inequalities to improved AM-GM inequality

Following an idea of Lin, we prove that if A and B are two positive operators such that 0 < mI ≤ A ≤ m′I ≤ M′I ≤ B ≤ MI, then

$${\Phi ^2}\left( {\frac{{A + B}}{2}} \right) \leqslant \frac{{{K^2}\left( h \right)}}{{{{\left( {1 + \frac{{{{\left( {\log \frac{{M'}}{{m'}}} \right)}^2}}}{8}} \right)}^2}}}{\Phi ^2}\left( {A\# B} \right),$$

and

$${\Phi ^2}\left( {\frac{{A + B}}{2}} \right) \leqslant \frac{{{K^2}\left( h \right)}}{{{{\left( {1 + \frac{{{{\left( {\log \frac{{M'}}{{m'}}} \right)}^2}}}{8}} \right)}^2}}}{\left( {\Phi \left( A \right)\# \Phi \left( B \right)} \right)^2},$$

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.