Conformal mappings and inequalities for algebraic polynomials. II
type: Item
instance of: scholarly article
title: Conformal mappings and inequalities for algebraic polynomials. II
zbMATH Open document ID: 1156.30005
author: V. N. Dubinin
published in: Journal of Mathematical Sciences (New York)
publication date: 2008-08-18T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 5312790
zbMATH Keywords: polynomials (one complex variable)
zbMATH Keywords: inequalities for sums
zbMATH Keywords: series and integrals of real functions
MaRDI profile type: Publication
full work available at URL: s10958-005-0318-5
OpenAlex ID: W1997397915
cites work: A sharp Schwarz inequality on the boundary
cites work: The Schwarz inequality on the boundary for functions regular in the disk
cites work: The boundary distortion and extremal problems in certain classes of univalent functions
cites work: Distortion theorems for polynomials on a circle
cites work: Application of conformal mappings to inequalities for polynomials
cites work: Q5549176
cites work: Inequalities concerning the derivative of polynomials
cites work: Q4347188
cites work: Q4000460
cites work: Q4305242
DOI: 10.1007/S10958-005-0318-5
Recommended article: Application of conformal mappings to inequalities for polynomials
Recommended article: Q4780014
Recommended article: Schwarz's lemma and estimates of coefficients for regular functions with free domain of definition
Recommended article: Estimates for the polar derivative of a constrained polynomial on a disk
Recommended article: Q3560373
Recommended article: Q3412515
Recommended article: On the distortion theorems for algebraic polynomials
Recommended article: Smirnov&#039;s inequality for polynomials having zeros outside the unit disc
Recommended article: Differential inequalities for algebraic polynomials
Recommended article: Inequalities for the polar derivative of a polynomial
description: scientific article; zbMATH DE number 5312790