  The purpose of this paper is twofold. Firstly, we will compute the explicit
expression of the Rawnsley's $\varepsilon$-function
$\varepsilon_{(\alpha,g(\mu;\nu))}$ of
$\big(\big(\prod_{j=1}^k\Omega_j\big)^{{\mathbb{B}}^{d_0}}(\mu),g(\mu;\nu)\big)$,
where $g(\mu;\nu)$ is a K\"ahler metric associated with the K\"ahler potential
$-\sum_{j=1}^k\nu_j\ln
N_{\Omega_j}(z_j,\overline{z_j})^{\mu_j}-\ln(\prod_{j=1}^kN_{\Omega_j}(z_j,\overline{z_j})^{\mu_j}-\|w\|^2)$
on the generalized Cartan-Hartogs domain
$\big(\prod_{j=1}^k\Omega_j\big)^{{\mathbb{B}}^{d_0}}(\mu)$ and obtain
necessary and sufficient conditions for $\varepsilon_{(\alpha,g(\mu;\nu))}$ to
become a polynomial in $1-\|\widetilde{w}\|^2$. Secondly, we study the Berezin
quantization on $\big(\prod_{j=1}^k\Omega_j\big)^{{\mathbb{B}}^{d_0}}(\mu)$
with the metric $ g(\mu;\nu)$.
