Some functional lower bounds for Fejér's sine polynomial

Let $\alpha\in\mathbb{R}$. We prove that \begin{align*} \sum_{k=1}^{n}\frac{\sin k\theta}{k}&\geqslant\left(\frac{335}{576}+\frac{31}{128}\alpha\right)\sin\theta+\left(\frac{39}{128}+\frac{5}{32}\alpha\right)\sin 2\theta+\left(\frac{113}{2304}+\frac{21}{256}\alpha\right)\sin 3\theta\\ &\quad\,+\left(\frac{25}{256}-\frac{5}{64}\alpha\right)\sin 4\theta+\left(\frac{125}{2304}-\frac{25}{256}\alpha\right)\sin 5\theta＞\theta^2\left(\cot\frac{\theta}{2}-\frac{\pi-\theta}{2}\right) \end{align*} for every integer $n\geqslant 2$ and $\theta\in(0,\pi)$ if and only if $$ \frac{\pi^2}{24}\leqslant\alpha\leqslant\alpha_0, $$ where $\alpha_0$ denotes the unique real zero of the cubic polynomial $$ 41721661440t^3-57761574336t^2+46230194016t-13479325596. $$ Both bounds for $\alpha$ are sharp. This sharpens a result due to Alzer and Koumandos.