# A pseudo Fejér-Jackson inequality problem

$x\in (0,\pi)$ ,Prove that: \begin{align} \sum_{k=1}^{n}\frac{\sin{kx}}{k}>x\left(1-\frac{x}{\pi}\right)^3 \end{align}

the inequality holds for all integer $n$

I tried Fourier, or Dirichlet kernel, but they don't work.Thanks for your attention!

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Please state where this problem from. –  nbubis Aug 2 '12 at 12:30
while proving the Fejér-Jackson inequality, from artofproblemsolving.com/Forum/viewtopic.php?t=114058 5 floor –  Golbez Aug 2 '12 at 12:32