New inequalities of Fejér—Jackson-type

The classical Fejér—Jackson inequality states that for $n\geq 0$ and $x\in[0,\pi]$, $$ \sum_{k=0}^n\frac{\sin((k+1)x)}{k+1}\geq 0. $$ Here, we present an extension and a counterpart of this result. We prove that the inequalities $$ \sum_{k=0}^n\frac{\sin((ck+1)x)}{ck+1}\geq 0\quad\mbox{ and }\quad\sum_{k=0}^n(-1)^k\frac{\sin((ck+1)x)}{ck+1}\geq 0 $$ are valid for all integers $c\geq 1$, $n\geq 0$, and real numbers $x\in[0,\pi]$.