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Inequalities for Alternating Trigonometric Sums |
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| Authors: | Horst Alzer Xiuping Liu Xiquan Shi |
| Institution: | 1. Morsbacher Str. 10, 51545, Waldbr?l, Germany 2. Department of Applied Mathematics, Dalian University of Technology, Dalian, 116024, China 3. Department of Applied Mathematics and Theoretical Physics, Delaware State University, Dover, DE, 19991, USA |
| Abstract: | In 1970, J.B. Kelly proved that $$\begin{array}{ll}0 \leq \sum\limits_{k=1}^n (-1)^{k+1} (n-k+1)|\sin(kx)| \quad{(n \in \mathbf{N}; \, x \in \mathbf{R})}.\end{array}$$ We generalize and complement this inequality. Moreover, we present sharp upper and lower bounds for the related sums $$\begin{array}{ll} & \sum\limits_{k=1}^{n} (-1)^{k+1}(n-k+1) | \cos(kx) | \quad {\rm and}\\ & \quad{\sum\limits_{k=1}^{n} (-1)^{k+1}(n-k+1)\bigl( | \sin(kx) | + | \cos(kx)| \bigr)}.\end{array}$$ |
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