On the sine polynomials of Fejér and Lukács

The sine polynomials of Fejér and Lukács are defined by \(\begin{aligned} F_n(x)=\sum _{k=1}^n\frac{\sin (kx)}{k} \quad \text{ and } \quad L_n(x)=\sum _{k=1}^n (n-k+1)\sin (kx), \end{aligned}\) $$\begin{array}{c}\hfill F_n(x)=\sum _{k=1}^n\frac{sin(kx)}{k}\text{and}{L}_n(x)=\sum _{k=1}^n(n-k+1)sin(kx),\end{array}$$ respectively. We prove that for all \(n\ge 2\) $n\ge 2$ and \(x\in (0,\pi )\) $x\in (0,\pi )$ , we have \(\begin{aligned} F_n(x)\le \lambda \, L_n(x) \quad \text{ and } \quad \mu \le \frac{1}{F_n(x)}-\frac{1}{L_n(x)} \end{aligned}\) $$\begin{array}{c}\hfill F_n(x)\le \lambda L_n(x)\text{and}\mu \le \frac{1}{F_n(x)}-\frac{1}{L_n(x)}\end{array}$$ with the best possible constants \(\begin{aligned} \lambda = \frac{8-3\sqrt{2}}{12(2-\sqrt{2})} \quad \text{ and } \quad \mu =\frac{2}{9}\sqrt{3}. \end{aligned}\) $$\begin{array}{c}\hfill \lambda =\frac{8-3\sqrt{2}}{12(2-\sqrt{2})}\text{and}\mu =\frac{2}{9}\sqrt{3}.\end{array}$$ An application of the first inequality leads to a class of absolutely monotonic functions involving the arctan function.