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Classes of Nonnegative Sine Polynomials

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Trigonometric Sums and Their Applications

Abstract

We present several one-parameter classes of nonnegative sine polynomials. One of our theorems states that the inequality

$$\displaystyle 0\leq \sum _{k=1}^n \Bigl (\frac {1}{n}+\frac {1}{k}\Bigr )(n-k+\alpha )\sin {}(kx) \quad {(\alpha \in \mathbb {R})} $$

holds for all n ≥ 1 and x ∈ [0, π] if and only if α ∈ [0, 3]. This extends a result of Dimitrov and Merlo (2002), who proved the inequality for α = 1.

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Authors and Affiliations

  1. Waldbröl, Germany

    Horst Alzer

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

    Man Kam Kwong

Authors
  1. Horst Alzer
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  2. Man Kam Kwong
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Correspondence to Horst Alzer .

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Editors and Affiliations

  1. Moscow Institute of Physics and Technology, Dolgoprudny, Russia, Moscow state University, Moscow, Russia, Buryat State University, Ulan-Ude, Russia, Caucasus Mathematical Center, Adyghe State University, Maykop, Russia

    Andrei Raigorodskii

  2. Institute of Mathematics, University of Zurich, Zurich, Switzerland, Moscow Institute of Physics and Technology, Dolgoprudny, Russia, Institute for Advanced Study Program in Interdisciplinary Studies, Princeton, NJ, USA

    Michael Th. Rassias

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Alzer, H., Kwong, M.K. (2020). Classes of Nonnegative Sine Polynomials. In: Raigorodskii, A., Rassias, M. (eds) Trigonometric Sums and Their Applications. Springer, Cham. https://doi.org/10.1007/978-3-030-37904-9_3

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