An extension of some inequalities of P. Erdős and P. Turán concerning algebraic polynomials having all real zeros
type: Item
instance of: scholarly article
title: An extension of some inequalities of P. Erdős and P. Turán concerning algebraic polynomials having all real zeros
zbMATH Open document ID: 0916.26004
DOI: 10.1007/BF00058939
author: B. Underhill
author: A. K. Varma
published in: Acta Mathematica Hungarica
publication date: 1999-05-18T00:00:00Z
review text: The authors consider inequalities of the following type \[ \int_{-1}^1 \rho(x) (P'_n(x))^pdx \leq K_n \int_{-1}^1 \rho(x) (P_n(x))^pdx \] where the weight function \(\rho:[-1,1]\to[0,\infty)\) and the power \(p>0\) are fixed and \(P_n\) is an arbitrary algebraic polynomial of degree at most \(n\) belonging to a prescribed class of polynomials. The problem is to find the constant \(K_n\) such that the above inequality is valid and sharp. The authors discuss the following cases in detail and obtain the best possible constant \(K_n\): (i) \(\rho(x)=(1-x^2)^\alpha\) (where \(\alpha>-1\)), \(p=2\), and \(P_n\in L_n\); (ii) \(\rho(x)=(1-x^2)^3\), \(p=4\), and \(P_n\in L_n\); (iii) \(\rho(x)=(1-x^2)^\alpha\) (where \(\alpha>-1\)), \(p=2\), and \(P_n\in H_n\); (iv) \(\rho(x)=(1-x^2)^3\), \(p=4\), and \(P_n\in H_n\); (v) \(\rho(x)=1\), \(p\geq 2\) even, and \(P_n\in H_n\), \noindent where \(L_n\) denotes the class of polynomials whose degree is \(n\), and \(P_n\in L_n\) is of the form \[ P_n(x)=\sum_{k=0}^na_k(1+x)^{n-k}(1-x)^k \qquad (a_k\geq 0), \] and \(H_n\) denotes the class of those polynomials whose degree is \(n\) and whose zeros are all real and lie inside \((-1,1)\).
reviewed by: Zsolt Páles
Mathematics Subject Classification ID: 26D05
zbMATH DE Number: 1288360
zbMATH Keywords: inequalities
zbMATH Keywords: algebraic polynomials
MaRDI profile type: Publication
cites work: On extremal properties of the derivatives of polynomials
cites work: An extremum problem concerning algebraic polynomials
cites work: On some generalizations of a theorem of A. Markoff
cites work: The degree of approximation by polynomials with positive coefficients
cites work: Extremal Problems for Lorentz Classes of Nonnegative Polynomials in L 2 Metric with Jacobi Weight
cites work: Some Inequalities of Algebraic Polynomials Having all Zeros Inside [ -1, 1 ]
cites work: Some Inequalities of Algebraic Polynomials having Real Zeros
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description: scientific article; zbMATH DE number 1288360