Monotonicity theorems and inequalities for certain sine sums

Inspired by the work of Askey-Steinig, Szeg\"o, and Schweitzer, we provide several monotonicity theorems and inequalities for certain sine sums. Among others, we prove that for $n\geq 1$ and $x\in (0,\pi/2)$, we have $$ \frac{d}{dx} \frac{C_n(x)}{1-\cos(x)}<0 \quad\mbox{and} \quad \frac{d}{dx} \left(1-\cos(x)\right)C_n(x)>0, $$ where $$ C_n(x)=\sum_{k=1}^n\frac{\sin((2k-1)x)}{2k-1} $$ denotes Carslaw's sine polynomial. Another result states that the inequality $$ \sum_{k=1}^n (n-k+a)(n-k+b) k \sin(kx)>0 \quad (a,b\in \mathbb{R}) $$ holds for all $n\geq 1$ and $x\in (0,\pi)$ if and only if $a=b=1$. Many corollaries and applications of these results are given. Among them, we present a two-parameter class of absolutely monotonic rational functions.