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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 ?

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    That is the Fejer-Jackson inequality. A bit of Googling will help for sure. – 2017-01-12
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    i think a proof was found by Vietoris an Austrian mathematician – 2017-01-12
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    Thank you. Just edited the OP. – 2017-01-14

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