On polynomials monotonic on the unit interval
type: Item
instance of: scholarly article
title: On polynomials monotonic on the unit interval
zbMATH Open document ID: 1017.30003
publication date: 2001-09-17T00:00:00Z
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 30C75
zbMATH DE Number: 1647682
zbMATH Keywords: bound for the derivative
zbMATH Keywords: monotonic polynomials
author: Mohammed A. Qazi
MaRDI profile type: Publication
full work available at URL: anly.2001.21.2.129
OpenAlex ID: W2333611136
DOI: 10.1524/ANLY.2001.21.2.129
published in: Analysis (München)
review text: The reviewed paper contains the proof of the following theorem 1. Let \(f\) be a polynomial of even degree \(n\) with \(|f(x)|\leq 1\) for \(x\in [-1,1]\). Besides, let \(f\) be monotonic on \([-1,1]\). Then NEWLINE\[NEWLINE |f^\prime(x)|\leq \tfrac{1}{4}n(n+2),\quad x\in [-1,1]\tag{\(*\)} NEWLINE\]NEWLINE The proof is a technical one, but rather complicated. The representation formula for monotonic polynomials is used. The extremal polynomial is determined and is given via Jacobi polynomials \(P^{(0,1)}(x).\) The corresponding bound in \((*)\) for odd degree polynomials was known earlier and is equal \(\frac{1}{4}(n+1)^2.\)
reviewed by: Jan Szynal
description: scientific article; zbMATH DE number 1647682