Tarnovetska O. Investigation of the classes of convergence of entire functions of several complex variables

In the thesis, the main object of research is the convergence classes of entire analytic functions of several complex variables in the Reinhardt domains. Such classes are quite wide. When studying the properties of such functions, classes of integrals convergence naturally arise from the characteristics of entire (or more general analytic) functions, in particular, in the theory of the distribution of values. G. Valiron (1923) first established the conditions of belonging of entire functions of finite order to the convergence class in terms of conditions on the Taylor coefficients of their power expansions. Subsequently, this result was repeatedly generalized and carried over to various classes of analytic functions represented by power series, Dirichlet series, or Laplace-Stieltjes integrals. Recently (2016) О.М. Mulyava and M.M. Sheremeta described in terms of Taylor coefficients, the conditions for the belonging of entire functions of several variables to convergence classes, which are determined by integrals of the maximum of the modulus of an entire function on the exhaustions of the space by full multiples circular domains. The exhaustions mentioned above must satisfy certain additional conditions that are not fulfilled, for example, for some model widely used exhaustions, such as exhaustion with balls.This significantly narrows the possible applicability of the results. In this regard, a natural topical problem arises -- to remove these additional restrictions on exhaustion. On the one hand, it is well known that each analytic function in the full Reinhardt domain can be represented in this domain by a multiple power series. On the other hand, the domain of convergence of each multiple power series is the logarithmically convex full Reinhardt domain.Therefore, it is relevant to consider for entire functions of many variables of the convergence class, determined on the basis of exhaustions by the full Reinhardt domains, as well as the corresponding analogues for analytic functions represented by multiple power series in these domains. Naturally, the question of the possibility of obtaining analogues of statements about belonging to the convergence classes for multiple Dirichlet series arises. The main results of the dissertation concern both these and some related problems.