Inequalities for the derivatives of polynomials with real zeros
type: Item
instance of: scholarly article
title: Inequalities for the derivatives of polynomials with real zeros
zbMATH Open document ID: 0641.42001
DOI: 10.1007/BF01057288
author: S. A. Pichugov
author: V. F. Babenko
published in: Ukrainian Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: The following sharp inequalities are established. Let \(P_ n\) be a polynomial of degree \(n\geq 2\) with all its zeros on \([-1,1]\). Then \(\| P_ n''\| \geq \min\{n,(n-1)n/4\}\| P_ n\|\) (sup norms). For a trigonometric polynomial \(T_ n\) of order n with 2n zeros on \([0,2\pi)\), the corresponding result is \(\| T_ n''\| \geq n\| T_ n\|.\) If also \(0<h<\min \{h(T_ n')\), \(2\cos^{-1}(1/2)^{{1/2}n}\}\), where (T) denotes the minimum distance between the zeros of T, then \[ \max_ x |\Delta^ 2_ h T_ n(x)\geq 2(1-(\cos h/2)^{2n})\| T_ n\|. \]
Mathematics Subject Classification ID: 42A05
Mathematics Subject Classification ID: 26C05
Mathematics Subject Classification ID: 26D05
zbMATH DE Number: 4044416
zbMATH Keywords: inequalities
zbMATH Keywords: trigonometric polynomial
MaRDI profile type: Publication
cites work: Q3663911
full work available at URL: https://doi.org/10.1007/bf01057288
OpenAlex ID: W1985457244
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description: scientific article; zbMATH DE number 4044416