Some inequalities for interpolational operator means

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 \[\Phi \left( A{{\sharp}_{\alpha }}B \right)\le \Phi \left( \left( A{{\sharp}_{\alpha }}B \right){{\sharp}_{\beta }}A \right){{\sharp}_{\alpha }}\Phi \left( \left( A{{\sharp}_{\alpha }}B \right){{\sharp}_{\beta }}B \right)\le \Phi \left( A \right){{\sharp}_{\alpha }}\Phi \left( B \right).\]