Petrechko N. Properties of functions of bounded index in the unit bidisc.

The dissertation consists of an introduction, 4 chapters, conclusions to each section and general conclusions, list of sources used. The introduction substantiates the relevance of the research topic, formulates the purpose, task, subject, object and methods of the research, presents the scientific novelty, the practical significance of the results obtained, the relationship of work with scientific themes and the personal contributions of the author of the dissertation, a list of conferences and scientific seminars, on which the results of the dissertation research are tested; List of publications in which the main results of the dissertation are published. In the dissertation, the main object of investigations is a class of analytic functions in the unit polydisc --- so-called functions of bounded $\mathbf{L}$-index in joint variables. There were obtained the criteria of boundedness of $\mathbf{L}$-index in joint variables for analytic functions in a polydisc. Particularly, the statements describe estimates of the behavior of the maximum modulus of the function on polydiscs of various radii, local behavior of the maximum modulus of partial derivatives on polydiscs of various radii. We found sufficient conditions of boundedness of the $\mathbf{L}$-index in joint variables for the solutions of some higher-order linear systems of partial differential equations with analytic coefficients in the polydisc. The properties of the power expansion for entire functions in complex plane and analytic functions in the unit bidisc of bounded $\mathbf{L}$-index in joint variables are investigated. {We also indicate growth estimates of logarithm of maximum modulus on a bidisc for this class of analytic functions. The logarithm behave as some integral from the vector-function $\mathbf{L}$ in the worst case. In any compact embedding domain in the unit bidisc an analytic functions in the bidisc has bounded $\mathbf{L}$-index in joint variables for every positive continuous vector-function $\mathbf{L}$, which is greater than some constant depending of the domain. All results of the thesis are new. They have theoretical meaning and can be used both in multidimensional complex analysis and in the analytic theory of differential equations. \textbf{Keywords:} entire function, analytic function, polydisc, bounded $\mathbf{L}$-index in joint variables, system of linear partial differential equations.