Some estimates for the Bergman kernel and metric in terms of logarithmic capacity

Abstract

For a bounded domain $\Omega$ on the plane we show the inequality $c_{\Omega}(z)^{2} \leq 2\pi K_{\Omega}(z)$, $z \in \Omega$, where $c_{\Omega}(z)$ is the logarithmic capacity of the complement $\mathbb{C} \setminus \Omega$ with respect to $z$ and $K_{\Omega}$ is the Bergman kernel. We thus improve a constant in an estimate due to T. Ohsawa but fall short of the inequality $c_{\Omega}(z)^{2} \leq \pi K_{\Omega}(z)$ conjectured by N. Suita. The main tool we use is a comparison, due to B. Berndtsson, of the kernels for the weighted complex Laplacian and the Green function. We also show a similar estimate for the Bergman metric and analogous results in several variables.