Inequalities for trigonometric sums

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