  We prove that a continuous function $f:(0,\infty) \to (0,\infty)$ is operator
monotone increasing if and only if $f(A \: !_t \: B) \leqs f(A) \: !_t \: f(B)$
for any positive operators $A,B$ and scalar $t \in [0,1]$. Here, $!_t$ denotes
the $t$-weighted harmonic mean. As a counterpart, $f$ is operator monotone
decreasing if and only if the reverse of preceding inequality holds. Moreover,
we obtain many characterizations of operator-monotone
increasingness/decreasingness in terms of operator means. These
characterizations lead to many operator inequalities involving means.
