Inequalities for trigonometric sums

TL;DR

This paper introduces new inequalities for trigonometric sums, refining classical results and establishing sharp bounds, with applications to absolutely monotonic functions.

Contribution

The paper presents novel inequalities for trigonometric sums, including a sharp bound that refines the classical Szeg"o-Schweitzer inequality.

Findings

Established a new inequality with a sharp constant factor 2/9.

Refined the classical positivity result for sine sums.

Derived a two-parameter class of absolutely monotonic functions.

Abstract

We present several new inequalities for trigonometric sums. Among others, we show that the inequality $$ \sum_{k=1}^n (n-k+1)(n-k+2)k\sin(kx) > \frac{2}{9} \sin(x) \bigl( 1+2\cos(x) \bigr)^2 $$ holds for all $n\geq 1$ and $x\in (0, 2\pi/3)$. The constant factor $2/9$ is sharp. This refines the classical Szeg\"o-Schweitzer inequality which states that the sine sum is positive for all $n\geq 1$ and $x\in (0,2 \pi/3)$. Moreover, as an application of one of our results, we obtain a two-parameter class of absolutely monotonic functions.