On some F\'ejer-type trigonometric sums

TL;DR

This paper investigates four types of Fejér-type trigonometric sums, analyzing their growth, spikes, and positivity, with a focus on the case where both functions are sine, supporting a conjecture about their positivity.

Contribution

It provides a detailed analysis of Fejér-type sums, especially the sine-sine case, and offers evidence supporting the conjecture that these sums remain positive.

Findings

Sums exhibit unbounded growth as n increases.

Graphs show spikes at certain x values, explained in the paper.

Strong evidence supports the conjecture that S_n(x)>0 for 0<x<π.

Abstract

We examine the four F\'ejer-type trigonometric sums of the form \[S_n(x)=\sum_{k=1}^n \frac{f(g(kx))}{k}\qquad (0<x<\pi)\] where $f(x)$, $g(x)$ are chosen to be either $\sin x$ or $\cos x$. The analysis of the sums with $f(x)=g(x)=\cos x$, $f(x)=\cos x$, $g(x)=\sin x$ and $f(x)=\sin x$, $g(x)=\cos x$ is reasonably straightforward. It is shown that these sums exhibit unbounded growth as $n\to\infty$ and also present `spikes' in their graphs at certain $x$ values for which we give an explanation. The main effort is devoted to the case $f(x)=g(x)=\sin x$, where we present arguments that strongly support the conjecture made by H. Alzer that $S_n(x)>0$ in $0<x<\pi$. The graph of the sum in this case presents a jump in the neighbourhood of $x=2\pi/3$. This jump is explained and is quantitatively estimated when $n\to\infty$.