We prove that
$$
\left|\sum_{k=1}^{n}\frac{\sin((2k-1)x)}{k}\right|
{\rm Si}(\pi)=1.8519\dots
$$
for all integers $n\geq 1$ and real numbers $x$. The upper bound
${\rm Si}(\pi)$ is best possible. This result refines inequalities due to
Fejér (1910) and Lenz (1951).
@article{10_4064_cm96_1_8,
author = {Horst Alzer and Stamatis Koumandos},
title = {A sharp bound for a sine polynomial},
journal = {Colloquium Mathematicum},
pages = {83--91},
year = {2003},
volume = {96},
number = {1},
doi = {10.4064/cm96-1-8},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm96-1-8/}
}
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AU - Horst Alzer
AU - Stamatis Koumandos
TI - A sharp bound for a sine polynomial
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Horst Alzer; Stamatis Koumandos. A sharp bound for a sine polynomial. Colloquium Mathematicum, Tome 96 (2003) no. 1, pp. 83-91. doi: 10.4064/cm96-1-8
