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Title: On a trigonometric inequality of askey and steinig
Authors: Alzer, H
Kwong, MK 
Keywords: Trigonometric sums
Issue Date: 2018
Publisher: IOS Press
Source: Asymptotic analysis, 2018, v. 106, no. 3-4, p. 233-249 How to cite?
Journal: Asymptotic analysis 
Abstract: In 1974, Askey and Steinig showed that for n >= 0 and x is an element of (0, 2 pi), S-n(x) = Sigma(n)(k=0) sin((k+1/4)x)/k+1 > 0 and C-n(x) = Sigma(n)(k=0) cos((k+1/4x)/k+1 > 0. We prove that S-n(x) + C-n(x) >= 1/root 2 and that the alternating sums S*(n)(x) = Sigma(n)(k=0) (-1)(k) sin((k+1/4)x)/k+1 and C*(n)(x) = Sigma(n)(k=0) (-1)(k) cos((k+1/4x)/k+1 satisfy S*(n)(x) + C*(n)(x) >= 1/200(13 - root 85) root 300+20 root 85 = 0.41601.... Both inequalities hold for all n >= 0 and x is an element of [0, 2 pi]. The constant lower bounds given in (0.1) and (0.2) are best possible. The asymptotic behaviour of both sums is also investigated.
ISSN: 0921-7134
EISSN: 1875-8576
DOI: 10.3233/ASY-171447
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