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On the trigonometric polynomials of Fejér and Young

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Abstract

The trigonometric polynomials of Fejér and Young are defined by \(S_n (x) = \sum\nolimits_{k = 1}^n {\tfrac{{\sin (kx)}} {k}}\) and \(C_n (x) = 1 + \sum\nolimits_{k = 1}^n {\tfrac{{\cos (kx)}} {k}}\), respectively. We prove that the inequality \(\left( {{1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-\nulldelimiterspace} 9}} \right)\sqrt {15} \leqslant {{C_n \left( x \right)} \mathord{\left/ {\vphantom {{C_n \left( x \right)} {S_n \left( x \right)}}} \right. \kern-\nulldelimiterspace} {S_n \left( x \right)}}\) holds for all n ≥ 2 and x ∈ (0, π). The lower bound is sharp.

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Authors and Affiliations

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

    Horst Alzer

  2. China University of Geosciences, Wuhan, P. R. China, 430074

    Qinghe Yin

Authors
  1. Horst Alzer
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  2. Qinghe Yin
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Correspondence to Horst Alzer.

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Communicated by András Kroó

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Alzer, H., Yin, Q. On the trigonometric polynomials of Fejér and Young. Period Math Hung 63, 81–87 (2011). https://doi.org/10.1007/s10998-011-7081-9

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  • DOI: https://doi.org/10.1007/s10998-011-7081-9

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