Inequalities for two sine polynomials

Horst Alzer¹ ; Stamatis Koumandos²

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

Colloquium Mathematicum, Tome 105 (2006) no. 1, pp. 127-134

Résumé

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 $x\in (0,\pi-\pi/n]$.

Détail

DOI : 10.4064/cm105-1-11

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