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Abstract:
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Dissertation is written in 60 pages and is divided into next parts:
1) Preface (pages 2-7)
2) Introduction (pages 8-29)
3) Concentration polynomial in low degrees (pages 30-56)
4) References (pages 57-60) which is consisted of 52 items
Chapter 2 is divided into 9, and chapter 3 into 2 sections. In preface a short historical review of polynomials and their importance and position in mathematics are given. Especially interesting parts in preface are about number of zeros of polynomials in different sections of complex plane. In section 2.1 there are well known characteristic of Möbijus’ transformation which will be used further in dissertation. Section 2.2 of same chapter is consisted of relations of different norms which are being introduced to vector spaces of all polynomials with complex coefficients. In section 2.3 Hurwitz polynomials are explained. This class of polynomials which was being examined at the end of 19th century has found its real position in subject which is being examined in this dissertation. Jensen's formula (which also appeared at the end of 19th century) is described in section 4 from more aspects. In sections 5, 6, 7 and 8 the relation among Jensen's formula, Hardy's spaces of p degree, generalized Jensen's formula and Mahler's measure is given. In section 9 in dissertation the story about lower and upper boundaries of Jensen's functional is given (definition, motivation, some well known results and some new results of the author). The chapter 3 is consisted of results of the author which are related to lower boundaries of Jensen's functional for polynomials which satisfy the condition (1.2) of dissertation. In that case extreme functions are being determined. The main purpose of author is making intervals [-2k,-2k log 2] whose ends presents asymptotically lower and upper boundary of best lower boundary of Jensen's functional determined. The part of those results is published in "Computers and Mathematics with Applications". |