  In this paper we address two problems concerning a family of domains
$M_{\Omega}(\mu) \subset \C^n$, called Cartan-Hartogs domains, endowed with a
natural Kaehler metric $g(\mu)$. The first one is determining when the metric
$g(\mu)$ is extremal (in the sense of Calabi), while the second one studies
when the coefficient $a_2$ in the Engli\v{s} expansion of Rawnsley
$\epsilon$-function associated to $g(\mu)$ is constant.
