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Title: On a sine polynomial of Turan
Authors: Alzer, H
Kwong, MK 
Keywords: Trigonometric sums
Chebyshev polynomials
Issue Date: 2018
Publisher: Rocky Mountain Mathematics Consortium
Source: Rocky mountain journal of mathematics, 2018, v. 48, no. 1, p. 1-18 How to cite?
Journal: Rocky mountain journal of mathematics 
Abstract: In 1935, Turan proved that S-n,S-a(x) = Sigma(n)(j=1) ((n+a-j)(n-j)) sin(jx) > 0, n, a is an element of N, 0 < x < pi. We present various related inequalities. Among others, we show that the refinements S-2n-1,S-a(x) >= sin(x) and S-2n,S-a (x) >= 2 sin(x)(1 + cos(x)) are valid for all integers n >= 1 and real numbers a >= 1 and x is an element of (0, pi). Moreover, we apply our theorems on sine sums to obtain inequalities for Chebyshev polynomials of the second kind.
ISSN: 0035-7596
DOI: 10.1216/RMJ-2018-48-1-1
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