On a sine polynomial of Turán. (English) Zbl 1392.26024

The authors prove inequalities for the trigonometric polynomials \[ \begin{aligned} S_{n,a}(x)&=\sum\limits_{j=1}^n \binom{n+a-j}{n-j}\sin(jx)\\ \Theta_{n,b}(x,y)&=\sum\limits_{j=1}^n \binom{n+b-j}{n-j}\frac{\sin(jx)\sin(jy)}{j} \end{aligned} \] where \(a\), \(b\) are real parameters, \(n\geq 1\), \(x,y\in(0,\pi).\) P. Turán proved the positivity of these polynomials if \(a\), \(b\) are natural numbers (see [J. Lond. Math. Soc. 10, 277–280 (1935; JFM 61.0278.02); ibid. 13, 278–282 (1938; Zbl 0020.01403)]). It is proved that each of these polynomials are positive for all \(n\geq 1\) and for all \(x,y\in(0,\pi)\) if and only if \(a\geq 1.\) Several lower bounds are found for these polynomials, among others \[ S_{2n-1,a}(x)\geq \sin x, \qquad S_{2n,a}(x)\geq 2\sin x(1+\cos x) \] if \(n\geq 1, a\geq 1\) and \(x\in(0,\pi),\) and \[ \Theta_{2n-1,b}(x,y)\geq \sin x\sin y,\qquad \Theta_{2n,b}(x,y)\geq 2\sin x\sin y(1+\cos x\cos y), \] provided that \(n\geq 1\), \(a\geq 1\) and \(x,y\in(0,\pi).\) Analogous inequalities are proved for the polynomials \(S_{n,a}^*,\Theta_{n,b}^*\) which are obtained from \(S_{n,a},\Theta_{n,b}\) by restricting the summations in them to odd \(j\)’s only. Finally, the results are applied to obtain inequalities for Chebyshev polynomials.


26D05 Inequalities for trigonometric functions and polynomials
26D15 Inequalities for sums, series and integrals
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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[1] H. Alzer and B. Fuglede, On a trigonometric inequality of Turán, J. Approx. Th. 164 (2012), 1496-1500. · Zbl 1253.42001
[2] H. Alzer and S. Koumandos, On the partial sums of a Fourier series, Constr. Approx. 27 (2008), 253-268. · Zbl 1152.42300
[3] H. Alzer and M.K. Kwong, Extension of a trigonometric inequality of Turán, Acta Sci. Math. 80 (2014), 21-26. · Zbl 1324.26022
[4] H.S. Carslaw, A trigonometrical sum and Gibbs phenomenon in Fourier’s series, Amer. J. Math. 39 (1917), 185-198. · JFM 46.0461.01
[5] L. Fejér, Einige Sätze, die sich auf das Vorzeichen einer ganzen rationalen Funktion beziehen;..., Monatsh. Math. Phys. 35 (1928), 305-344. · JFM 54.0314.03
[6] T.H. Gronwall, Über die Gibbssche Erscheinung und die trigonometrischen Summen \(\sin x +\frac{1}{2} \sin 2x +\cdots +\frac{1}{n} \sin nx\), Math. Ann. 72 (1912), 228-243. · JFM 43.0323.01
[7] D. Jackson, Über eine trigonometrische Summe, Rend. Circ. Mat. Palermo 32 (1911), 257-262. · JFM 42.0281.03
[8] J.C. Mason and D.C. Handscomb, Chebyshev polynomials, Chapman and Hall, Boca Raton, 2002. · Zbl 1015.33001
[9] G.V. Milovanović, D.S. Mitrinović and Th.M. Rassias, Topics in polynomials: Extremal problems, inequalities, zeros, World Scientific, Singapore, 1994. · Zbl 0848.26001
[10] J. Steinig, A criterion for the positivity of sine polynomials, Proc. Amer. Math. Soc. 38 (1973), 583-586. · Zbl 0256.42002
[11] G. Szegő, Power series with multiply sequences of coefficients, Duke Math. J. 8 (1941), 559-564. · JFM 67.0257.04
[12] P. Turán, Über die arithmetischen Mittel der Fourierreihe, J. Lond. Math. Soc. 10 (1935), 277-280. · JFM 61.0278.02
[13] —-, Über die Partialsummen der Fourierreihe, J. Lond. Math. Soc. 13 (1938), 278-282. · Zbl 0020.01403
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