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.
Citation
Zbigniew Błocki. "Some estimates for the Bergman kernel and metric in terms of logarithmic capacity." Nagoya Math. J. 185 143 - 150, 2007.
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