Sharp Bounds for a Sine Sum

We prove the double-inequality \[ \frac{r(1-r^n)}{1+r}<\sum_{k=1}^n\frac{r^k}{k}\frac{\sin(kx)}{\sin(x)} < \frac{r(1-r^n)}{1-r}. \] The left-hand side holds for even \(n\geqslant 2\), \(r\in (0,1]\), \(x\in (0,\pi)\), whereas the right-hand side is valid for \(n\geqslant 2\), \(r\in (0,1)\), \(x\in (0,\pi)\). Both bounds are sharp. An application of the second inequality leads to a new two-parameter class of absolutely monotonic functions.