We prove: (I) For all integers $n\geq 2$ and real numbers
$x\in (0,\pi)$ we have
$$
\alpha \leq \sum_{j=1}^{n-1}\frac{1}{n^2-j^2} \sin(jx) \leq \beta,
$$
with the best possible constant bounds
$$
\alpha=\frac{15-\sqrt{2073}}{10240}\sqrt{1998-10\sqrt{2073}}=
-0.1171\dots
,\quad\ \beta=\frac{1}{3}.
$$
(II) The inequality
$$
0\sum_{j=1}^{n-1}{(n^2-j^2)} \sin(jx)
$$
holds for all even integers $n\geq 2$ and $x\in (0,\pi)$, and also for all odd
integers $n\geq 3$ and $x\in (0,\pi-\pi/n]$.
Keywords:
prove integers geq real numbers have alpha leq sum n frac j sin leq beta best possible constant bounds alpha frac sqrt sqrt sqrt dots quad beta frac inequality sum n j sin holds even integers geq odd integers geq pi
Affiliations des auteurs :
Horst Alzer
1
;
Stamatis Koumandos
2
1
Morsbacher Str. 10 D-51545 Waldbröl, Germany
2
Department of Mathematics and Statistics The University of Cyprus P.O. Box 20537 1678 Nicosia, Cyprus
@article{10_4064_cm105_1_11,
author = {Horst Alzer and Stamatis Koumandos},
title = {Inequalities for two sine polynomials},
journal = {Colloquium Mathematicum},
pages = {127--134},
year = {2006},
volume = {105},
number = {1},
doi = {10.4064/cm105-1-11},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm105-1-11/}
}
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AU - Horst Alzer
AU - Stamatis Koumandos
TI - Inequalities for two sine polynomials
JO - Colloquium Mathematicum
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Horst Alzer; Stamatis Koumandos. Inequalities for two sine polynomials. Colloquium Mathematicum, Tome 105 (2006) no. 1, pp. 127-134. doi: 10.4064/cm105-1-11
