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Published in:
2020 | OriginalPaper | Chapter
Inequalities for Weighted Trigonometric Sums
Authors : Horst Alzer, Omran Kouba
Published in: Trigonometric Sums and Their Applications
Publisher: Springer International Publishing
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Abstract
We prove that the double-inequality holds for all even integers n ≥ 2 and positive real numbers w
j (j = 1, …, n) with w
1 + ⋯ + w
n = 1 if and only if a ≤ 1 and b ≥ 2. Moreover, we present a cosine counterpart of this result.
$$\displaystyle \left ( \sum _{j=1}^n \frac {w_j}{ 1-\sin ^2 \frac {j\pi }{n+1} } \right )^a \leq \sum _{j=1}^n \frac {w_j}{ 1-\sin \frac {j\pi }{n+1} } \cdot \sum _{j=1}^n \frac {w_j}{ 1+\sin \frac {j\pi }{n+1} } \leq \left ( \sum _{j{=}1}^n \frac {w_j}{ 1{-}\sin ^2 \frac {j\pi }{n{+}1} } \right )^b $$