We show that the (weighted) harmonic operator mean is characterized 
as an operator mean $m$ satisfying $F(A m B)\le F(A) m F(B)$
for every operator monotone function $F$ on $(0,\infty)$ 
based on the numerical means. 
We also show the non-affine representing function $f_m(x)=1\,m\,x$ 
of an operator mean $m$ is an extreme point of the set 
of representing functions $F$ with $F\circ f_m\leqq f_m\circ F$.