Sketch of the Theory of Growth of Holomorphic Functions in a Multidimensional Torus

Abstract

We develop an approach to the theory of growth of the class H(????ⁿ) of holomorphic functions in a multidimensional torus ????ⁿ based on the structure of elements of this class and well-known results of the heory of growth of entire functions of several complex variables. This approach is illustrated in the case where the growth of the function g ∈ H(????ⁿ) is compared with the growth of its maximum modulus on the skeleton of the polydisk. The properties of the corresponding characteristics of growth of the functions in the class H(????ⁿ) are studied with their relation to coefficients of the corresponding Laurent series. A comparative analysis of these results and similar assertions of the theory of growth of entire functions of several variables is given.