Log-hyperconvexity index and Bergman kernel

We obtain a quantitative estimate of Bergman distance when $\mathrm{\Omega }\subset {\mathbb{C}}^n$ is a bounded domain with log-hyperconvexity index ${\alpha }_l(\mathrm{\Omega })>\frac{n-1+\sqrt{(n-1)(n+3)}}{2}$, as well as the $A^2(\log A{)}^q$-integrability of the Bergman kernel $K_{\mathrm{\Omega }}(\cdot ,w)$ when ${\alpha }_l(\mathrm{\Omega })>0$.