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

## TL;DR

This paper provides detailed proofs and technical lemmas for the sine inequality involving a weighted sine polynomial, including computational methods, to establish non-negativity over a specific interval.

## Contribution

It offers a comprehensive, detailed proof of the sine inequality, with auxiliary lemmas and computational techniques, extending prior results and clarifying the proof structure.

## Key findings

- The sine polynomial is nonnegative for specified parameters.
- Several lemmas of independent interest are proved.
- Computational methods assist in verifying complex inequalities.

## 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\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.