A new bound for the Fejér-Jackson sum
type: Item
instance of: scholarly article
title: A new bound for the Fejér-Jackson sum
zbMATH Open document ID: 0912.42001
DOI: 10.1023/A:1006560422934
author: Gavin Brown
author: Stamatis Koumandos
published in: Acta Mathematica Hungarica
publication date: 1999-03-14T00:00:00Z
review text: A positive lower estimate is given here for the sum \[ S_n (\vartheta) =\sum^n_{k=1} \frac {\sin k \vartheta}{k}. \] The inequality \[ S_n(\vartheta)>\frac{1-\sin \frac\vartheta 2}{\cos \frac \vartheta 2} \tag{1} \] holds for \(\vartheta_n<\vartheta <\pi\) if \(n\) is odd and \(\vartheta_n<\vartheta \leq \pi-\frac{2\pi}{2n+1}\) if \(n\) is even, where \(\vartheta_n\) is the unique zero of \(S_n(\vartheta)-\frac{1-\sin \frac\vartheta 2}{\cos \frac \vartheta 2}\) in the interval \((0,\frac{\pi}{n+1})\). Using (1) the authors give the proof for an earlier result of \textit{R. Askey} and \textit{J. Steining} [Am. J. Math. 98, 357-365 (1976; Zbl 0334.42003)] according to which \[ \frac{d}{d\vartheta} \left(\sum^n_{k=1} \frac{\sin k\vartheta}{k\sin \frac{\vartheta}{2}}\right) <0 \qquad \text{for}\quad \vartheta \in (0, \pi). \]
Mathematics Subject Classification ID: 42A05
zbMATH DE Number: 1261205
zbMATH Keywords: Fejér-Jackson sum
reviewed by: Tibor Šalát
MaRDI profile type: Publication
full work available at URL: a:1006560422934
OpenAlex ID: W2060821519
description: scientific article; zbMATH DE number 1261205