Rocky Mountain Journal of Mathematics

On a sine polynomial of Turán

Horst Alzer and Man Kam Kwong

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In 1935, Tur\'an proved that \[ S_{n,a}(x)= \sum _{j=1}^n{n+a-j\choose n-j} \sin (jx)>0, \] \[n,a\in \mathbf {N},\quad 0\lt x\lt \pi .\] We present various related inequalities. Among others, we show that the refinements $$ S_{2n-1,a}(x)\geq \sin (x) \quad \mbox {and} \quad {S_{2n,a}(x)\geq 2\sin (x)(1+\cos (x))} $$ are valid for all integers $n\geq 1$ and real numbers $a\geq 1$ and $x\in (0,\pi )$. Moreover, we apply our theorems on sine sums to obtain inequalities for Chebyshev polynomials of the second kind.

Article information

Rocky Mountain J. Math., Volume 48, Number 1 (2018), 1-18.

First available in Project Euclid: 28 April 2018

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Zentralblatt MATH identifier

Primary: 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.) [See also 42C05 for general orthogonal polynomials and functions]

Trigonometric sums inequalities Chebyshev poly­no­mi­als


Alzer, Horst; Kwong, Man Kam. On a sine polynomial of Turán. Rocky Mountain J. Math. 48 (2018), no. 1, 1--18. doi:10.1216/RMJ-2018-48-1-1.

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