On the sine polynomials of Fejér and Lukács

IF 0.5 4区 数学 Q3 MATHEMATICS
Horst Alzer, Man Kam Kwong
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引用次数: 0

Abstract

The sine polynomials of Fejér and Lukács are defined by

$$\begin{aligned} F_n(x)=\sum _{k=1}^n\frac{\sin (kx)}{k} \quad \text{ and } \quad L_n(x)=\sum _{k=1}^n (n-k+1)\sin (kx), \end{aligned}$$

respectively. We prove that for all \(n\ge 2\) and \(x\in (0,\pi )\), we have

$$\begin{aligned} F_n(x)\le \lambda \, L_n(x) \quad \text{ and } \quad \mu \le \frac{1}{F_n(x)}-\frac{1}{L_n(x)} \end{aligned}$$

with the best possible constants

$$\begin{aligned} \lambda = \frac{8-3\sqrt{2}}{12(2-\sqrt{2})} \quad \text{ and } \quad \mu =\frac{2}{9}\sqrt{3}. \end{aligned}$$

An application of the first inequality leads to a class of absolutely monotonic functions involving the arctan function.

论费耶尔和卢卡奇的正弦多项式
费耶尔和卢卡奇的正弦多项式定义如下: $$begin{aligned}F_n(x)=sum _{k=1}^n\frac\sin (kx)}{k}\quad \text{ and }\L_n(x)=sum _{k=1}^n (n-k+1)\sin (kx),end{aligned}$$。我们证明,对于所有的(nge 2)和(xin (0,\pi)),我们有$$\begin{aligned}。F_n(x)/le /lambda /, L_n(x) /quad /text{ and }\quad \mu \le \frac{1}{F_n(x)}-\frac{1}{L_n(x)} \end{aligned}$$ 有最好的常数 $$\begin{aligned}\λ = (frac{8-3/sqrt{2}}{12(2-/sqrt{2})}\quad \text{ and }\quad \mu =\frac{2}{9}\sqrt{3}.\end{aligned}$$应用第一个不等式可以得到一类涉及 arctan 函数的绝对单调函数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Archiv der Mathematik
Archiv der Mathematik 数学-数学
CiteScore
1.10
自引率
0.00%
发文量
117
审稿时长
4-8 weeks
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.
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