Convexity of the Bergman Kernels on Convex Domains

Let $\Omega$ be a convex domain in $\mathbb{C}^n$ and $\varphi$ a convex function on $\Omega$.

We prove that $\log{K_{\Omega,\varphi}(z)}$ is a convex function (might be identically $-\infty$) on $\Omega$, where $K_{\Omega,\varphi}$ is the weighted Bergman kernel.

When $\varphi\equiv0$, we prove a Brunn-Minkowski type inequality, which further implies that $K_\Omega(z)^{-\frac{1}{2n}}$ is a convex function if $\Omega$ is convex.

Some necessary and sufficient conditions for strictly convexity are also obtained.