Convexity of the Bergman Kernels on Convex Domains

Let $\mathrm{\Omega }$ be a convex domain in ${\mathbb{C}}^n$ and $\phi$ a convex function on $\mathrm{\Omega }$. We prove that $\log K_{\mathrm{\Omega },\phi }(z)$ is a convex function (might be identically $-\mathrm{\infty }$) on $\mathrm{\Omega }$, where $K_{\mathrm{\Omega },\phi }$ is the weighted Bergman kernel. When $\phi \equiv 0$, we prove a Brunn-Minkowski type inequality, which further implies that $K_{\mathrm{\Omega }}(z{)}^{-\frac{1}{2n}}$ is a convex function if $\mathrm{\Omega }$ is convex. Some necessary and sufficient conditions for strictly convexity are also obtained.