Characterizations of Operator Monotonicity via Operator Means and Applications to Operator Inequalities

We prove that a continuous function $f:(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ is operator monotone increasing if and only if $f(A{!}_{t}B)\text{leqs}f(A){!}_{t}f(B)$ 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.