Inequalities for trigonometric sums

We present several new inequalities for trigonometric sums. Among others, we show that the inequality \(\begin{aligned} \sum _{k=1}^n (n-k+1)(n-k+2)k\sin (kx) > \frac{2}{9} \sin (x) \bigl ( 1+2\cos (x) \bigr )^2 \end{aligned}\) $$\begin{array}{c}\hfill \sum _{k=1}^n(n-k+1)(n-k+2)ksin(kx)>\frac{2}{9}sin(x)(1+2cos(x){)}^2\end{array}$$ holds for all \(n\ge 1\) $n\ge 1$ and \(x\in (0, 2\pi /3)\) $x\in (0,2\pi /3)$ . The constant factor 2/9 is sharp. This refines the classical Szegö-Schweitzer inequality which states that the sine sum is positive for all \(n\ge 1\) $n\ge 1$ and \(x\in (0,2 \pi /3)\) $x\in (0,2\pi /3)$ . Moreover, as an application of one of our results we obtain a two-parameter class of absolutely monotonic functions.