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We generalize the inequality being a counterpart of the several complexvariables version of the Suita conjecture. For this aim higher ordergeneralizations of the Bergman kernel are introduced. As a corollary some newpartial results on the dimension of the Bergman space in pseudoconvex domainsare given. A relation between the problem of Wiegerinck on possible dimensionof the Bergman space of unbounded pseudoconvex domains in general case and inthe case of balanced domains is also shown. Moreover, some classes of domainswhere the answer to the problem of Wiegerinck is positive are given.Additionally, regularity properties of functions involving the volumes ofAzukawa indicatrices are shown.

Generalizations of the higher dimensional Suita conjecture and its relation with a problem of Wiegerinck