  Consider the mean field equation with critical parameter $8\pi$ in a bounded
smooth domain $\Omega$. Denote by $E_{8\pi}(\Omega)$ the infimum of the
associated functional $I_{8\pi}(\Omega)$. We call $E_{8\pi}(\Omega)$ the
"energy" of the domain $\Omega$. We prove that if the area of $\Omega$ is equal
to $\pi$, then the energy of $\Omega$ is always greater or equal to the energy
of the unit disk and equality holds if and only if $\Omega$ is the unit disk.
We also give a sufficient condition for the existence of a minimizer for
$I_{8\pi}(\Omega)$.
