Extremal properties of derivative of algebraic polynomials
type: Item
instance of: scholarly article
title: Extremal properties of derivative of algebraic polynomials
zbMATH Open document ID: 0817.26006
DOI: 10.1007/BF01874697
author: J. Burkett
author: A. K. Varma
published in: Acta Mathematica Hungarica
publication date: 1995-07-18T00:00:00Z
review text: A problem of P. Turán runs: If \(\phi(x)\geq 0\) in \(I= [-1,1]\), let \(C\) be the class of all polynomials \(p\) of degree at most \(n\), such that \(| p(x)|\leq \phi(x)\) in \(I\). How large can \(\max_ I | p^{(k)}(x)|\) be if \(p\in C\)? The authors obtained some weighted \(L^ 2\)-inequalities in a similar spirit. For example, if \(p\) is a real algebraic polynomial of degree \((n+1)\), such that \(| p(x)|\leq (1- x^ 2)^{1/2}\) for \(x\in I\), then for \(k= 2,3,\dots\) we have \[ \int^ 1_{-1} | p^{(k)}(x)|^ 2(1- x^ 2)^{(2k- 3)/2} dx\leq \int^ 1_{-1} | f^{(k)}(x)|^ 2(1- x^ 2)^{(2k- 3)/2} dx, \] where \(f(x)= (1- x)^ 2 u_{n-1}(x)\) and \(u_ n\) denotes the Chebyshev polynomial of the second kind. Equality occurs if and only if \(p(x)= \pm f(x)\).
Mathematics Subject Classification ID: 26C05
zbMATH DE Number: 638212
zbMATH Keywords: range
zbMATH Keywords: extremal property
zbMATH Keywords: real algebraic polynomial
zbMATH Keywords: Chebyshev polynomial
reviewed by: I. N. Baker
MaRDI profile type: Publication
cites work: On a Problem of Turan About Polynomials
cites work: On a Problem of Turán about Polynomials II
cites work: On a Problem of Turan About Polynomials with Curved Majorants
cites work: Some recent results on the flow of Non-Newtonian fluids
cites work: Q4026707
cites work: Inequalities for trigonometric polynomials
cites work: Markoff type inequalities for curved majorants in the weighted \(L^ 2\) norm
cites work: Q4840254
Recommended article: On extremal properties of the derivatives of polynomials
Recommended article: Q3637961
Recommended article: On Extremal Properties of the Derivatives of Polynomials and Rational Functions
Recommended article: Extremal Properties of Polynomials
Recommended article: Q4888688
Recommended article: On some extremal properties of algebraic polynomials
Recommended article: On extremal properties for the polar derivative of polynomials
Recommended article: Extremal properties of logarithmic derivatives of polynomials
Recommended article: Extremal problems for algebraic polynomials
description: scientific article; zbMATH DE number 638212