Based on the Kubo-Ando theory of operator means, 
we give a proof of the well-known L\"owner-Heinz 
theorem which asserts that for bounded linear 
operators $A$ and $B$, if $A \ge B \ge 0$ then 
$A^p \ge B^p$ for $0 \le p \le 1$. A key fact 
for the proof of the theorem is its special case 
for $p = 1/2$: if $A \ge B \ge 0$ then $A^{1/2} 
\ge B^{1/2}$, which says that the geometric mean 
$X^{1/2}$ of the identity operator $1$ and a positive 
operator $X$ is monotone. We give a short proof of 
this fact, using the arithmetic-harmonic mean defined 
by J. I. Fujii.