Inequalities for Weighted Trigonometric Sums

Horst Alzer () and Omran Kouba ()
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Omran Kouba: Higher Institute for Applied Sciences and Technology, Department of Mathematics

A chapter in Trigonometric Sums and Their Applications, 2020, pp 85-96 from Springer

Abstract: Abstract We prove that the double-inequality ∑ j = 1 n w j 1 − sin 2 j π n + 1 a ≤ ∑ j = 1 n w j 1 − sin j π n + 1 ⋅ ∑ j = 1 n w j 1 + sin j π n + 1 ≤ ∑ j = 1 n w j 1 − sin 2 j π n + 1 b $$\displaystyle \left ( \sum _{j=1}^n \frac {w_j}{ 1-\sin ^2 \frac {j\pi }{n+1} } \right )^a \leq \sum _{j=1}^n \frac {w_j}{ 1-\sin \frac {j\pi }{n+1} } \cdot \sum _{j=1}^n \frac {w_j}{ 1+\sin \frac {j\pi }{n+1} } \leq \left ( \sum _{j{=}1}^n \frac {w_j}{ 1{-}\sin ^2 \frac {j\pi }{n{+}1} } \right )^b $$ 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.

Keywords: Inequalities; Sine sums; Cosine sums; Optimal constants (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/978-3-030-37904-9_4

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