  In this paper we investigate the balanced condition (in the sense of
Donaldson) and the existence of an Englis expansion for the LeBrun's metrics on
$C^2$. Our first result shows that a LeBrun's metric on $C^2$ is never balanced
unless it is the flat metric. The second one shows that an Englis expansion of
the Rawnsley's function associated to a LeBrun's metric always exists, while
the coefficient $a_3$ of the expansion vanishes if and only if the LeBrun's
metric is indeed the flat one.
