A refinement of the discrete Wirtinger inequality
type: Item
instance of: scholarly article
title: A refinement of the discrete Wirtinger inequality
zbMATH Open document ID: 0854.26011
author: Xin-Min Zhang
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-01-19T00:00:00Z
review text: In the present paper, the following refinement of Tang's discrete Wirtinger inequality [\textit{D. Tang}, Bull. Aust. Math. Soc. 43, No. 3, 467-474 (1991; Zbl 0751.26006)] is established: Theorem A. Let \(f(\theta)\) be a positive \(C^2\)-function on \((0,\ell)\) such that \(f'(\theta) f''(\theta) \neq 0\) and \(f'(\theta)^2 - f(\theta)f''(\theta) = \mu\), for \(\theta \in (0,\ell)\), where \(\mu\) is a constant. Then, if \(f''(\theta) < 0\) (or \(> 0)\)) on \((0,\ell)\), we have \[ \left( \sum^n_{i = 1} f(\theta_i)\right)^2 - c_n \sum^n_{i = 1} f(\theta_i) f'(\theta_i) \geq [\text{or } \leq] \left( nf(\sigma) - \sum^n_{i = 1} f(\theta_i)\right)^2 \] where \(\theta_i \in (0,\ell)\) with \(\theta_1 + \dots + \theta_n = m\ell\), \(\sigma = m\ell/n\), \(c_n = nf(\sigma)/f'(\sigma)\), and \(m(\leq n)\) is a positive constant. Equality in the last inequality holds if and only if \(\theta_1 = \dots = \theta_n = \sigma\). Two notable consequences of Theorem A for the cases when \(m =2\), \(\mu = 1\), \(\ell = \pi/2\), and \(f(\theta) = \sin \theta\) and \(f(\theta) = \cos \theta\) are derived in the Theorems B and \(\text{B}'\), respectively. As a geometric application of Theorem A, a Benneson-style isoperimetric inequality for plane polygons is obtained in Theorem C. The classification of all the functions verifying Theorem \(A\) is given. Finally, the main result (Theorem A) is restated as a weighted exponential inequality.
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 908599
zbMATH Keywords: discrete Wirtinger inequality
zbMATH Keywords: isoperimetric inequality
zbMATH Keywords: plane polygons
zbMATH Keywords: weighted exponential inequality
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0232
OpenAlex ID: W1969215793
DOI: 10.1006/JMAA.1996.0232
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description: scientific article; zbMATH DE number 908599