Lower bounds on the Bergman metric near points of infinite type

Let $\mathrm{\Omega }$ be a pseudoconvex domain in ${\mathbb{C}}^n$ satisfying an $f$-property for some function $f$. We show that the Bergman metric associated to $\mathrm{\Omega }$ has the lower bound $\tilde{g}({\delta }_{\mathrm{\Omega }}(z{)}^{-1})$ where ${\delta }_{\mathrm{\Omega }}(z)$ is the distance from $z$ to the boundary $\mathrm{\partial }\mathrm{\Omega }$ and $\tilde{g}$ is a specific function defined by $f$. This refines Khanh-Zampieri’s work in \cite{KZ12} with reducing the smoothness assumption of the boundary.