Inequalities for two sine polynomials
Tom 105 / 2006
Colloquium Mathematicum 105 (2006), 127-134
MSC: 26D05, 42A05.
DOI: 10.4064/cm105-1-11
Streszczenie
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 $x\in (0,\pi-\pi/n]$.
