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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H. Alzer and S. Koumandos, Companions of the inequalities of Fejér-Jackson and Young, Anal. Math., 31 (2005), 75–84.
H. Alzer and S. Koumandos, A new refinement of Young’s inequality, Proc. Edinb. Math. Soc., 50 (2007), 255–262.
R. Askey and G. Gasper, Inequalities for polynomials, The Bieberbach conjecture (eds. A. Baernstein II, D. Drasin, P. Duren, A. Marden), Math. Surveys and Monographs, no. 21, Amer. Math. Soc., Providence, RI, 1986, 7–32.
D. Jackson, Über eine trigonometrische Summe, Rend. Circ. Mat. Palermo, 32 (1911), 257–262.
K. Knopp, Theorie und Anwendung der unendlichen Reihen, Springer, Berlin, 1964.
G. V. Milovanović, D. S. Mitrinović and Th. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Sci. Publ., Singapore, 1994.
B. L. Van Der Waerden, Algebra I, Springer, Berlin, 1971.
W. H. Young, On a certain series of Fourier, Proc. London Math. Soc. (2), 11 (1913), 357–366.
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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