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Abstract

We study the behavior of the pluricomplex Green function on a bounded hyperconvex domain $D$ that admits a smooth plurisubharmonic exhaustion function $\psi$ such that $1/|\psi|$ is integrable near the boundary of $D$, and moreover satisfies the estimate $|\psi | \leq C \exp ( - C' (\log (1/\delta_D)) ^\alpha )$ at points close enough to the boundary with constants $C,C'>0$ and $0<\alpha<1$. Furthermore, we obtain a Hopf lemma for such a function $\psi$. Finally, we prove a lower bound on the Bergman distance on $D$.