  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\geq 2, \, \beta \geq \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.
