Sharp inequalities for trigonometric sums

Abstract

We prove the following two theorems:

(I) Let $n \geqslant 1$ be a (fixed) integer. Then we have for $\theta \in (0, \pi)$: \[ \sum\limits^n_{k=1}\frac{\cos (k\theta)}{k}\leqslant -\log\left(\sin\left(\frac{\theta}{2}\right)\right)+\frac{\pi-\theta}{2}+\sigma_n, \] with the best possible constant $\sigma_n = \sum\nolimits^n_{k=1}(-1)^k/k$.

(II) For even integers $n \geqslant 2$ and for $\theta \in (0, \pi)$ we have \[ \sum\limits^n_{k=1}\frac{\sin(k\theta)}{k}\leqslant\alpha(\pi-\theta), \] with the best possible constant $\alpha = 0.66 395\ldots$.

Our results refine inequalities due to C. Hyltén-Cavallius ‘11’ and P. Turán ‘23’, respectively.