On an Inequality Related to a Certain Fourier Cosine Series

Abstract

We prove that the Fourier cosine series ${\sum }_{k=1}^{\infty }(-1{)}^{k+1}\frac{r^k\cos k\varphi }{k+2}$ assumes its maximum value at $\varphi =0$ for $\varphi \in [0,\pi )$ 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.

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