On Bernstein's inequality for ultraspherical polynomials
type: Item
instance of: scholarly article
title: On Bernstein's inequality for ultraspherical polynomials
zbMATH Open document ID: 0912.33004
author: Horst Alzer
published in: Archiv der Mathematik
publication date: 1999-05-10T00:00:00Z
review text: The author offers a proof for \[ (\sin t)^{s}| P_{n}^{(s)}(\cos t)| < \frac{ 2^{1-s}}{\Gamma(s)} \frac{ \Gamma(n+(3s/2))}{\Gamma(n+1+(s/2))}, \] where \(0<t<\pi\), \(0<s<1\), and \(P_{n}^{(s)}\) is an ultraspherical polynomial. This improves several previous results, including that of \textit{A. Laforgia} [Boll. Unione Mat. Ital., VII. Ser., A 6, 267-269 (1992; Zbl 0582.33009)], who had \(\Gamma(n+2) / \Gamma(n+1)\) as second factor on the right and whose ``elegant methods motivated'' the present paper.
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 26D07
zbMATH DE Number: 1145416
zbMATH Keywords: Bernstein inequality
zbMATH Keywords: gamma functions
zbMATH Keywords: ultraspherical polynomials
reviewed by: János Aczél
MaRDI profile type: Publication
full work available at URL: s000130050150
OpenAlex ID: W2052883893
DOI: 10.1007/S000130050150
description: scientific article; zbMATH DE number 1145416