Bergman functions on weakly uniformly perfect domains

We construct two classes of Zalcman-type domains, on which the Bergman distance functions exhibit certain pre-described boundary behaviors. Such examples also lead to generalizations of uniform perfectness in the sense of Pommerenke. These weakly uniformly perfect conditions can be characterized in terms of the logarithm capacity. We obtain lower estimates for the boundary behaviors of Bergman kernel functions on such domains.