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Using the logarithmic capacity, we give quantitative estimates of the Greenfunction, as well as lower bounds of the Bergman kernel for boundedpseudoconvex domains in $\mathbb C^n$ and the Bergman distance for boundedplanar domains. In particular, it is shown that the Bergman kernel satisfies$K_\Omega(z)\gtrsim \delta_\Omega(z)^{-2}$ for any bounded pseudoconvex domainwith $C^0-$boundary. An application to holomorphic motions is given.

Capacities, Green function and Bergman functions