  We study the Euler-Lagrange equation for several natural functionals defined
on a conformal class of almost Hermitian metrics, whose expression involves the
Lee form $\theta$ of the metric. We show that the Gauduchon metrics are the
unique extremal metrics of the functional corresponding to the norm of the
codifferential of the Lee form. We prove that on compact complex surfaces, in
every conformal class there exists a unique metric, up to multiplication by a
constant, which is extremal for the functional given by the $L^2$-norm of
$dJ\theta$, where $J$ denotes the complex structure. These extremal metrics are
not the Gauduchon metrics in general, hence we extend their definition to any
dimension and show that they give unique representatives, up to constant
multiples, of any conformal class of almost Hermitian metrics.
