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)$.
Categories: math.CV