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Remarks on a sine polynomial

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Abstract

Let n ≥ 0 be an integer. Then we have for \({x\in(0,\pi)}\) :

$$\sum_{k=0}^n { 2n+1 \choose n-k }\frac{\sin((2k+1)x)}{2k+1}\leq\frac{8^n \, n!}{(2n+1)!!}.$$

The upper bound is best possible. This complements a result of Fejér, who proved that the sine polynomial is positive on (0, π).

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Authors and Affiliations

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

    Horst Alzer

  2. Department of Mathematics and Statistics, The University of Cyprus, P.O. Box 20537, 1678, Nicosia, Cyprus

    Stamatis Koumandos

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  1. Horst Alzer
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  2. Stamatis Koumandos
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Correspondence to Stamatis Koumandos.

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Alzer, H., Koumandos, S. Remarks on a sine polynomial. Arch. Math. 93, 475–479 (2009). https://doi.org/10.1007/s00013-009-0055-y

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  • DOI: https://doi.org/10.1007/s00013-009-0055-y

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