Technical Details of the Proof of the Sine Inequality \\[1.2ex] {\normalsize $\sum _{k=1}^{n-1}{\left(\frac{n}{k}-\frac{k}{n}\right)}^{\beta }\sin (kx)\ge 0$

Abstract

In a recent study, H. Alzer and the author showed that the sine polynomial $$\sum _{k=1}^{n-1}{\left(\frac{n}{k}-\frac{k}{n}\right)}^{\beta }\sin (kx)>0$$ is nonnegative for $x\in [0,\pi ]$, $n\ge 2,\beta \ge {\beta }_1:=\frac{\log (2)}{\log (16/5)}.$ This result, among others, will be presented in a forthcoming article. The proof relies on quite a number of technical Lemmas and inequalities. We have decided to delegate all the tedious details of the proofs of these Lemmas in a separate article, namely, the current one. Some of the proofs require brute-force numerical computation, performed with the help of the computer software MAPLE. A few of the Lemmas included here are of independent interest.