Hi, I need an estimation for the following integral

$\int_{0}^{2\pi} K_N^4(s) ds $

where $K_N(s)= \frac{1}{N2\pi} (\frac{sin(Ns/2)}{sin(s/2)})$, the Fejer kernel.

I don't know how to obtain an estimation better than

$\int_{0}^{2\pi} K_N^4(s) ds < N^4$

Does anyone know a better estimation or some trigonometric tricks that can help me to improve my estimation?

Thanks in advance 


Imma