Estimates of invariant metrics on pseudoconvex domains with comparable Levi form

Abstract

Let $\Omega$ be a smoothly bounded pseudoconvex domain in $\mathbb{C}^{n}$ and let $z_{0} \in b\Omega$ be a point of finite type. We also assume that the Levi form of $b\Omega$ is comparable in a neighborhood of $z_{0}$. Then we get a quantity which bounds from above and below the Bergman metric, Caratheodory metric and Kobayashi metric in a small constant and large constant sense.