Rawnsley's $\varepsilon$-function on some Hartogs type domains over bounded symmetric domains and its applications

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 $g(\mu;\nu)$.