Ovchar I. Theorems of Wiman - Valiron's type for entire Dirichlet series with non-monotonic sequence of exponents.

The thesis consists of a list of denotations, introduction, 3 chapters divided into sections, conclusions and a bibliography, which includes 202 items. In Chapter 1 we introduce an overview of works related to Wiman's inequalities, Borel's relations for entire Dirichlet series (absolutely convergent in the whole complex plane ) with monotonic exponents, growth of entire functions in horizontal strips, asymptotic relations for Laplace-Stieltjes integrals and formulate the main results of the thesis.In Chapter 2 a new Wiman - Valiron type theorem about estimate of a general term of entire Dirichlet series with arbitrary exponents by means of its maximal term is proved. Based on this theorem, sufficient conditions for asymptotic equality for logarithms of maximum of the modulus of its sum and a maximal term of that series (Borel's relation) are obtained. Moreover, new theorems about generalization on the latter relation, some analogues theorems for entire Dirichlet series with monotonic exponents, a new Wiman's inequality for this class of functions and its exactness is proved. Theorems obtained in this chapter, are applied for investigation of regular growth of entire Dirichlet series with arbitrary exponents in horizontal strips, and for investigation of Dirichlet series with arbitrary complex exponents. The main idea in these investigations is that in sufficient conditions for some asymptotic relations for entire Dirichlet series with arbitrary exponents its exponents can be replaced by logarithms of modulus' of its coefficients. This idea is confirmed in scientific researches written by B.V.Vynnytskiy, M.M.Cheremeta, O.B.Skaskiv and might be used for investigations of analytic and periodic functions.In Chapter 3 some problems regarding Laplace - Stieltjes integrals with large positive and small negative parameters are being investigated. On proving theorems of this chapter some ideas based on probability methods, i.e. Chebyshov's inequality, and statements from Chapter 2, are used. It is proved in this chapter that the conditions for Wiman's inequality for Laplace - Stieltjes integrals with large positive parameters, recently established by O.B.Skaskiv and A.O.Kuryliak, are very close to necessarily ones. For Laplace - Stieltjes integrals defined on sufficient conditions for Wiman's inequality are proved. For this class of integrals conditions, that turned out to be sufficient outside of some exceptional set, for estimation of the integral by means of the maximum of its integrand on a carrier of the measure are found.