Proof of Kuhn's polynomial conjectures
type: Item
instance of: scholarly article
title: Proof of Kuhn's polynomial conjectures
zbMATH Open document ID: 0536.26010
DOI: 10.1007/BF01193783
published in: Archiv der Mathematik
publication date: 1985-01-01T00:00:00Z
review text: Let \(f\) be a polynomial of exact degree n mapping the interval \(I=[-1,1]\) into itself and having all roots in \(I\). The rational function \(R_ f(x) = \frac{1-f^ 2(x)}{(1-x^ 2)(f'(x))^ 2}\) is positive on those subintervals of \(I\) where it is regular, and its positive squareroot \(\sqrt{R_ f}\) satisfies \(\left(\sqrt{R_ f}\right)''>0\) there, unless either \(f\) or \(-f\) equals the Chebyshev polynomial \(T_ n\) (in these cases \(R_ f\) is constant and minimal). There is a similar result for trigonometric polynomials.
Mathematics Subject Classification ID: 26C10
Mathematics Subject Classification ID: 26C05
Mathematics Subject Classification ID: 26C15
zbMATH DE Number: 3851430
zbMATH Keywords: Kuhn's polynomial conjectures
zbMATH Keywords: algebraic and trigonometric polynomials with only real roots
zbMATH Keywords: polynomial inequalities
Wikidata QID: Q122956361
author: Gundorph K. Kristiansen
MaRDI profile type: Publication
cites work: Über eine Vermutung von P. Erdős
cites work: Some Inequalities for Algebraic and Trigonometric Polynomials
cites work: Characterization of polynomials by means of their stationary values
full work available at URL: https://doi.org/10.1007/bf01193783
OpenAlex ID: W2001047946
description: scientific article; zbMATH DE number 3851430