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