Conformal mappings and inequalities for algebraic polynomials. II
In the first part [Algebra Anal. 13, No. 5, 16--43 (2001; Zbl 1016.30004)], a new approach to deriving inequalities for algebraic polynomials was suggested. This is based on constructing, for a given polynomial, a conformal and univalent mapping of a certain class, which is followed by applying certain results of the geometric theory of functions of a complex variable to the class in question. Several methods for constructing conformal mappings were considered, and some inequalities were obtained, which, in terms of the theory of functions, can be treated as covering and distortion theorems for algebraic polynomials. The present paper supplements the previous paper. An analog of the Schwarz boundary lemma is proved for non-univalent regular mappings of subsets of the unit disk onto a disk. Based on this result, certain strengthened inequalities of Bernstein type for algebraic polynomials are obtained. The generalized Mendeleev problem is discussed. Two-sided bounds for the module of the derivative of a polynomial with critical points on an interval are established. Bounds for the coefficients of polynomials under certain constraints are provided.
scientific article; zbMATH DE number 5312790