Inequalities for two sine polynomials
Inequalities for two sine polynomials
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{