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

Alzer, Horst; Kwong, Man Kam

Rocky Mt. J. Math. 48, No. 1, 1-18 (2018).

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.

Reviewer: László Losonczi (Debrecen)

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

References:

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