On an Inequality Related to a Certain Fourier Cosine Series

TL;DR

This paper proves that a specific Fourier cosine series reaches its maximum at zero for all angles in [0, π) when the parameter r is in (0, 1], providing a more concise proof using Chebyshev polynomial generating functions.

Contribution

It offers a more compact proof of an existing inequality involving Fourier cosine series, utilizing Chebyshev polynomial generating functions.

Findings

The series attains its maximum at φ=0 for all r in (0,1].

The proof is simplified compared to previous methods.

The approach leverages Chebyshev polynomial generating functions.

Abstract

We prove that the Fourier cosine series $\sum_{k=1}^{\infty}(-1)^{k+1}\frac{r^k\cos{k\phi}}{k+2}$ assumes its maximum value at $\phi = 0$ for $\phi \in [0, \pi)$ regardless of $r$ if $r \in (0, 1]$. This was first proved by Arias de Reyna and van de Lune. The more compact proof presented here is based on a generating function of the Chebyshev Polynomials.