{"title":"On the sine polynomials of Fejér and Lukács","authors":"Horst Alzer, Man Kam Kwong","doi":"10.1007/s00013-023-01950-2","DOIUrl":null,"url":null,"abstract":"<div><p>The sine polynomials of Fejér and Lukács are defined by </p><div><div><span>$$\\begin{aligned} F_n(x)=\\sum _{k=1}^n\\frac{\\sin (kx)}{k} \\quad \\text{ and } \\quad L_n(x)=\\sum _{k=1}^n (n-k+1)\\sin (kx), \\end{aligned}$$</span></div></div><p>respectively. We prove that for all <span>\\(n\\ge 2\\)</span> and <span>\\(x\\in (0,\\pi )\\)</span>, we have </p><div><div><span>$$\\begin{aligned} F_n(x)\\le \\lambda \\, L_n(x) \\quad \\text{ and } \\quad \\mu \\le \\frac{1}{F_n(x)}-\\frac{1}{L_n(x)} \\end{aligned}$$</span></div></div><p>with the best possible constants </p><div><div><span>$$\\begin{aligned} \\lambda = \\frac{8-3\\sqrt{2}}{12(2-\\sqrt{2})} \\quad \\text{ and } \\quad \\mu =\\frac{2}{9}\\sqrt{3}. \\end{aligned}$$</span></div></div><p>An application of the first inequality leads to a class of absolutely monotonic functions involving the arctan function.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archiv der Mathematik","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00013-023-01950-2","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The sine polynomials of Fejér and Lukács are defined by
期刊介绍:
Archiv der Mathematik (AdM) publishes short high quality research papers in every area of mathematics which are not overly technical in nature and addressed to a broad readership.