Abstract
The inequality \(\sum\nolimits_{k = 1}^n {(n - k + a)(n - k + b)} \sin (kx)\cos (ky) >0(a,b \in \mathbb{R})\) is proved to hold for all n ∈ ℕ and x, y, ∈ ℝ with 0 < x + y < π, 0 < x - y < π if and only if 0 < ab ≤ a + b + 1. This extends a theorem of Turán, who showed that our inequality is valid for a = 1, b = 2, y = 0.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. E. Andrews, R. Askey and R. Roy, Special Functions, Camb. Univ. Press, Cambridge, 1999.
R. Askey, Orthogonal Polynomials and Special Functions, Reg. Conf. Ser. Appl. Math. 21, SIAM, Philadelphia, PA, 1975.
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 21, Amer. Math. Soc., Providence, RI, 1986, 7–32.
L. Fejér, Einige Sätze, die sich auf das Vorzeichen einer ganzen rationalen Funktion beziehen, Monatsh. Math. Phys., 35 (1928), 305–344.
G. V. Milovanović, D. S. Mitrinović and Th. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Sci. Publ., Singapore, 1994.
G. Szegő, Power series with multiply monotonic sequences of coefficients, Duke Math. J., 8 (1941), 559–564.
P. Turán, Über die arithmetischen Mittel der Fourierreihe, J. London Math. Soc., 10 (1935), 277–280.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by V. Totik
Supported by the Hong Kong Government GRF Grant PolyU 5012/10P and the Hong Kong Polytechnic University Grants G-YK49 and G-U751.
Rights and permissions
About this article
Cite this article
Alzer, H., Kwong, M.K. Extension of a trigonometric inequality of Turän. ActaSci.Math. 80, 21–26 (2014). https://doi.org/10.14232/actasm-013-562-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.14232/actasm-013-562-6