Moscow Mathematical Journal

Volume 20, Issue 3, July–September 2020  pp. 441–451.

A Generalization of the Fejér–Jackson Inequality and Related Results

Authors:  Horst Alzer (1) and Man Kam Kwong (2)
Author institution:(1) Morsbacher Straße 10, 51545 Waldbröl, Germany
(2) Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Hong Kong

Summary:

We present several results for trigonometric sums related to the classical Fejér–Jackson inequality, namely, $$0< \sum_{k=1}^n\frac{\sin(kx)}{k} \quad (n\geq 1,\, 0< x<\pi).$$ Among these are:

1. Let $r\in \mathbb{R}$. Then, $0< \sum\limits_{\substack{k=1 \\ k \,\text{odd}}}^n \frac{\sin(kx)}{k} \, r^k$ holds for all $n\geq 1$ and $x\in (0,\pi)$ if and only if $r\in (0,1]$.

2. Let $a\in\mathbb{R}$. Then, $0< \sum\limits_{k=0}^{n-1} \cos(kx) \biggl( \sum\limits_{j=k+1}^n {j\choose k} \frac{\sin((j-k)x)}{j} \, a^j \biggr)$ holds for all $n\geq 1$ and $x\in (0,\pi)$ if and only if $a\in (0,1/2]$. For $a=1/2$, the result reduces to that of Fejér–Jackson.

3. Let $b\in \mathbb{R}$. Then, $0< \sum\limits_{k=0}^{n-1} \cos(kx) \biggl( \sum\limits_{\substack{j=k+1 \\ j \,\text{odd}}}^n {j\choose k} \frac{\sin((j-k)x)}{j} \, b^j \biggr)$ holds for all $n\geq 1$ and $x\in (0,\pi)$ if and only if $b\in (0,1/2]$. An analogous result holds when “odd” is replaced by “even” and $(0,\pi )$ by $(0,\frac{\pi }{2} )$.

2010 Math. Subj. Class. 26D05, 33B10, 05A19

Keywords: Fejér–Jackson inequality, trigonometric sums, harmonic numbers, combinatorial identity.