A generalization of the Fejér–Jackson inequality and related results

We present several results for trigonometric sums related to the classical Fejér–Jackson inequality, namely,
$$ \displaystyle 0<\sum_{k=1}^n\frac{\sin(kx)}{k} {(n\geq 1, 0<x<\pi)}. $$
Among these are:
1. Let $r\in \mathbb{R}$. Then, $ 0<\sum\limits_{\substack{k=1 \\ k \text{odd}}}^n \frac{\sin(kx)}{k} r^k $ holds for all $n\geq 1$ and $x\in (0,\pi)$ if and only if $r\in (0,1]$.
2. Let $a\in\mathbb{R}$. Then, $ 0<\sum\limits_{k=0}^{n-1} \cos(kx) \biggl( \sum\limits_{j=k+1}^n {j\choose k} \frac{\sin((j-k)x)}{j} a^j \biggr) $ holds for all $n\geq 1$ and $x\in (0,\pi)$ if and only if $a\in (0,1/2]$. For $ a=1/2 $, the result reduces to that of Fejér–Jackson.
3. Let $b\in \mathbb{R}$. Then, $ 0< \sum\limits_{k=0}^{n-1} \cos(kx) \biggl( \sum\limits_{\substack{j=k+1 \\ j \text{odd}}}^n {j\choose k} \frac{\sin((j-k)x)}{j} b^j \biggr) $ holds for all $n\geq 1$ and $x\in (0,\pi)$ if and only if $b\in (0,1/2]$. An analogous result holds when “odd” is replaced by “even” and $ (0,\pi ) $ by $ (0,\frac{\pi }{2} ) $.