  We generalize the inequality being a counterpart of the several complex
variables version of the Suita conjecture. For this aim higher order
generalizations of the Bergman kernel are introduced. As a corollary some new
partial results on the dimension of the Bergman space in pseudoconvex domains
are given. A relation between the problem of Wiegerinck on possible dimension
of the Bergman space of unbounded pseudoconvex domains in general case and in
the case of balanced domains is also shown. Moreover, some classes of domains
where the answer to the problem of Wiegerinck is positive are given.
Additionally, regularity properties of functions involving the volumes of
Azukawa indicatrices are shown.
