I would like to prove that $$\sum_{k=1}^n\frac{\sin(kx)}{k}\ge 0$$ for all $n\in\mathbb{N}^\star$ and $x\in[0,\pi]$
EDIT : Thanks for the references. The result of Leopold Vietoris gives a nice extension of the Fejer-Jackson's inequality. I could not find more than the statement of his theorem ... Does anyone know where to find it (or at least a sketch of it) ?
I also found (somewhere on the net and without any proof) the following assertion (which seems true, according to a few plottings ...)
For all $n\in\mathbb{N}^\star$ and $x\in(0,\pi)$, we have :
$$\sum_{k=1}^n\frac{\sin(kx)}{k}\ge x\left(1-\frac{x}{\pi}\right)^3$$
If true, this result would give yet another proof of FJ's inequality. Any hints ?
