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Inequalities for Alternating Trigonometric Sums

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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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Authors and Affiliations

  1. Morsbacher Str. 10, 51545, Waldbröl, Germany

    Horst Alzer

  2. Department of Applied Mathematics, Dalian University of Technology, Dalian, 116024, China

    Xiuping Liu

  3. Department of Applied Mathematics and Theoretical Physics, Delaware State University, Dover, DE, 19991, USA

    Xiquan Shi

Authors
  1. Horst Alzer
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  2. Xiuping Liu
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  3. Xiquan Shi
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Correspondence to Horst Alzer.

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Alzer, H., Liu, X. & Shi, X. Inequalities for Alternating Trigonometric Sums. Results. Math. 63, 1215–1223 (2013). https://doi.org/10.1007/s00025-012-0264-8

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  • DOI: https://doi.org/10.1007/s00025-012-0264-8

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