On the sine polynomials of Fejér and Lukács,Archiv der Mathematik

当前位置： X-MOL 学术 › Arch. Math. › 论文详情

Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)

On the sine polynomials of Fejér and Lukács
Archiv der Mathematik ( IF 0.6 ) Pub Date : 2024-01-24 , DOI: 10.1007/s00013-023-01950-2
Horst Alzer , Man Kam Kwong

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.

更新日期：2024-01-25

点击分享查看原文

点击收藏

阅读更多本刊最新论文本刊介绍/投稿指南

全部期刊列表>>