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2018 On a sine polynomial of Turán
Horst Alzer, Man Kam Kwong
Rocky Mountain J. Math. 48(1): 1-18 (2018). DOI: 10.1216/RMJ-2018-48-1-1

Abstract

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.

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Horst Alzer. Man Kam Kwong. "On a sine polynomial of Turán." Rocky Mountain J. Math. 48 (1) 1 - 18, 2018. https://doi.org/10.1216/RMJ-2018-48-1-1

Information

Published: 2018
First available in Project Euclid: 28 April 2018

zbMATH: 06866697
MathSciNet: MR3795730
Digital Object Identifier: 10.1216/RMJ-2018-48-1-1

Subjects:
Primary: 26D05, 26D15, 33C45

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 1 • 2018
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