Abstract
We prove several new inequalities for trigonometric sums in two variables. One of our results states that the double-inequality
\begin{align} -\frac{2}{3}(\sqrt{2}-1) &\leq \sum_{k=1}^{n}\frac{\cos((k-1/2)x)\sin((k-1/2)y)}{k-1/2}\leq 2 \end{align}
holds for all integers $n\geq 1$ and real numbers $x,y \in [0,\pi]$. Both bounds are best possible.
Citation
Horst Alzer. Stamatis Koumandos. "Sharp inequalities for trigonometric sums in two variables." Illinois J. Math. 48 (3) 887 - 907, Fall 2004. https://doi.org/10.1215/ijm/1258131058
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