A functional bound for Young's cosine polynomial

Abstract

We prove that

$$\begin{aligned} \frac{5}{6} + \sum_{k=1}^{n} \frac{\cos k \theta}{k} \ge \frac{1}{4} (1+\cos \theta )^2 \quad (n=2,3,\ldots;\ \theta \in (0, \pi )), \end{aligned}$$

where equality holds if and only if $n = 2$ and $\theta = \pi - \cos^{-1} \frac{1}{3}$. This refines a result of Brown and Koumandos.