## 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.

### MSC:

 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.)

### Keywords:

trigonometric sums; inequalities; Chebyshev polynomials

### Citations:

Zbl 0020.01403; JFM 61.0278.02
Full Text:

### References:

  H. Alzer and B. Fuglede, On a trigonometric inequality of Turán, J. Approx. Th. 164 (2012), 1496-1500. · Zbl 1253.42001  H. Alzer and S. Koumandos, On the partial sums of a Fourier series, Constr. Approx. 27 (2008), 253-268. · Zbl 1152.42300  H. Alzer and M.K. Kwong, Extension of a trigonometric inequality of Turán, Acta Sci. Math. 80 (2014), 21-26. · Zbl 1324.26022  H.S. Carslaw, A trigonometrical sum and Gibbs phenomenon in Fourier’s series, Amer. J. Math. 39 (1917), 185-198. · JFM 46.0461.01  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  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  D. Jackson, Über eine trigonometrische Summe, Rend. Circ. Mat. Palermo 32 (1911), 257-262. · JFM 42.0281.03  J.C. Mason and D.C. Handscomb, Chebyshev polynomials, Chapman and Hall, Boca Raton, 2002. · Zbl 1015.33001  G.V. Milovanović, D.S. Mitrinović and Th.M. Rassias, Topics in polynomials: Extremal problems, inequalities, zeros, World Scientific, Singapore, 1994. · Zbl 0848.26001  J. Steinig, A criterion for the positivity of sine polynomials, Proc. Amer. Math. Soc. 38 (1973), 583-586. · Zbl 0256.42002  G. Szegő, Power series with multiply sequences of coefficients, Duke Math. J. 8 (1941), 559-564. · JFM 67.0257.04  P. Turán, Über die arithmetischen Mittel der Fourierreihe, J. Lond. Math. Soc. 10 (1935), 277-280. · JFM 61.0278.02  —-, Über die Partialsummen der Fourierreihe, J. Lond. Math. Soc. 13 (1938), 278-282. · Zbl 0020.01403
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