Technical Details of the Proof of the Sine Inequality \\[1.2ex] {\normalsize $\displaystyle \sum_{k=1}^{n-1}\left( \frac{n}{k} - \frac{k}{n} \right) ^β\sin(kx) \geq 0$

英文摘要：

In a recent study, H. Alzer and the author showed that the sine polynomial $$ \sum_{k=1}^{n-1} \left( \frac{n}{k} - \frac{k}{n} \right) ^β\,\sin(kx) > 0 $$ is nonnegative for $ x\in[0,π] $, $ n\geq 2, \, β\geq β_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.