Behavior of the Bergman kernel and metric near convex boundary points

The boundary behavior of the Bergman metric near a convex boundary point $z_0$ of a pseudoconvex domain $D\subset\mathbb{C}^n$ is studied. It turns out that the Bergman metric at points $z\in D$ in the direction of a fixed vector $X_0\in\mathbb{C}^n$ tends to infinity, when $z$ is approaching $z_0$, if and only if the boundary of $D$ does not contain any analytic disc through $z_0$ in the direction of $X_0$.