A new bound for the Fejér-Jackson sum (Q1280285)

A positive lower estimate is given here for the sum \[ S_n (\vartheta) =\sum^n_{k=1} \frac {\sin k \vartheta}{k}. \] The inequality \[ S_n(\vartheta)>\frac{1-\sin \frac\vartheta 2}{\cos \frac \vartheta 2} \tag{1} \] holds for \(\vartheta_n<\vartheta <\pi\) if \(n\) is odd and \(\vartheta_n<\vartheta \leq \pi-\frac{2\pi}{2n+1}\) if \(n\) is even, where \(\vartheta_n\) is the unique zero of \(S_n(\vartheta)-\frac{1-\sin \frac\vartheta 2}{\cos \frac \vartheta 2}\) in the interval \((0,\frac{\pi}{n+1})\). Using (1) the authors give the proof for an earlier result of \textit{R. Askey} and \textit{J. Steining} [Am. J. Math. 98, 357-365 (1976; Zbl 0334.42003)] according to which \[ \frac{d}{d\vartheta} \left(\sum^n_{k=1} \frac{\sin k\vartheta}{k\sin \frac{\vartheta}{2}}\right) <0 \qquad \text{for}\quad \vartheta \in (0, \pi). \]