  Using the properties of geometric mean, we shall show for any $0\le \alpha
,\beta \le 1$, \[f\left( A{{\nabla }_{\alpha }}B \right)\le f\left( \left(
A{{\nabla }_{\alpha }}B \right){{\nabla }_{\beta }}A \right){{\sharp}_{\alpha
}}f\left( \left( A{{\nabla }_{\alpha }}B \right){{\nabla }_{\beta }}B
\right)\le f\left( A \right){{\sharp}_{\alpha }}f\left( B \right)\] whenever
$f$ is a non-negative operator log-convex function, $A,B\in \mathcal{B}\left(
\mathcal{H} \right)$ are positive operators, and $0\le \alpha ,\beta \le 1$. As
an application of this operator mean inequality, we present several refinements
of the Aujla subadditive inequality for operator monotone decreasing functions.
  Also, in a similar way, we consider some inequalities of Ando's type. Among
other things, it is shown that if $\Phi $ is a positive linear map, then \[\Phi
\left( A{{\sharp}_{\alpha }}B \right)\le \Phi \left( \left( A{{\sharp}_{\alpha
}}B \right){{\sharp}_{\beta }}A \right){{\sharp}_{\alpha }}\Phi \left( \left(
A{{\sharp}_{\alpha }}B \right){{\sharp}_{\beta }}B \right)\le \Phi \left( A
\right){{\sharp}_{\alpha }}\Phi \left( B \right).\]

Functional Analysis