  If $\sigma$ is a symmetric mean and $f$ is an operator monotone function on
$[0, \infty)$, then $$f(2(A^{-1}+B^{-1})^{-1})\le f(A\sigma B)\le f((A+B)/2).$$
Conversely, Ando and Hiai showed that if $f$ is a function that satisfies
either one of these inequalities for all positive operators $A$ and $B$ and a
symmetric mean different than the arithmetic and the harmonic mean, then the
function is operator monotone.
  In this paper, we show that the arithmetic and the harmonic means can be
replaced by the geometric mean to obtain similar characterizations. Moreover,
we give characterizations of operator monotone functions using self-adjoint
means and general means subject to a constraint due to Kubo and Ando.
