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arXiv · 2307.04464

Inequalities for trigonometric sums

Horst Alzer·Man Kam Kwong
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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.

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Horst Alzer, Man Kam Kwong. 2023-07-10. Inequalities for trigonometric sums. https://arxiv.org/abs/2307.04464

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