A positive functional bound for certain sine sums (Q5932663)

Consider the sine sums of the form \(S_{r} (n,t)=\sum_{k=1}^{n}\sin kt/k+r\). The classical Fejér-Jackson inequality means that \(S_{0} (n,t)>0\). A result of \textit{G. Brown and D. C. Wilson} [Math. Ann. 285, No. 1, 57-74 (1989; Zbl 0663.42002)] shows that \(S_{1} (n,t)>0\). These two inequalities hold for all \(n\) and \(0<t<\pi\). The last result is sharpened in this paper by proving that \(S_{1} (n,t)>1-\sin \frac t 2/4\cos \frac t 2\) for \(t\) in an interval which depends on \(n\).