  The Cartan-Hartogs domains are defined as a class of Hartogs type domains
over irreducible bounded symmetric domains. For a Cartan-Hartogs domain
$\Omega^{B}(\mu)$ endowed with the natural K\"{a}hler metric $g(\mu),$ Zedda
conjectured that the coefficient $a_2$ of the Rawnsley's $\varepsilon$-function
expansion for the Cartan-Hartogs domain $(\Omega^{B}(\mu), g(\mu))$ is constant
on $\Omega^{B}(\mu)$ if and only if $(\Omega^{B}(\mu), g(\mu))$ is
biholomorphically isometric to the complex hyperbolic space. In this paper,
following Zedda's argument, we give a geometric proof of the Zedda's conjecture
by computing the curvature tensors of the Cartan-Hartogs domain
$(\Omega^{B}(\mu), g(\mu))$.
