Convexity of the Bergman Kernels on Convex Domains

Abstract

Let $Ω$ be a convex domain in $\mathbb{C}^n$ and $φ$ a convex function on $Ω$. We prove that $\log{K_{Ω,φ}(z)}$ is a convex function (might be identically $-\infty$) on $Ω$, where $K_{Ω,φ}$ is the weighted Bergman kernel. When $φ\equiv0$, we prove a Brunn-Minkowski type inequality, which further implies that $K_Ω(z)^{-\frac{1}{2n}}$ is a convex function if $Ω$ is convex. Some necessary and sufficient conditions for strictly convexity are also obtained.