I have the following question: Let f be in $\mathbf{L}_{\mathbf{R}}^2([-\pi;\pi])$. Show that $$\left({\int_{-\pi}^\pi |x^nf(x)|\,\mathrm{d}\lambda(x)} \right ) \leq \frac{2*\pi^{2n+1}}{2n+1} {\int_{-\pi}^\pi |f|^2\,\mathrm{d}\lambda(x)}$$
Ok I have tried to use the inequality by letting $f = |\sum_{k=-\infty}^{\infty}c_ke^{ikx}| \leq \sum_{k=-\infty}^{\infty}|c_k|$ but after some calculations where I basicly take the this sum out and integrate $|x^n|$ this gives that the initial formula is less than or equal to $(\sum_{k=-\infty}^{\infty}|c_k|)^2\frac{4*\pi^{2n+2}}{n^2}$. This I can't show is lesser than or equal to what is above. I shouldn't have get rid of the $e^{ikx}$ thing to use the orthogonal relationship to get just $(\sum_{k=-\infty}^{\infty}|c_k|^2)$ and then use the Parseval equation to prove what I was supposed to. However I'm finding it difficult to not exclude $e^{ikx}$ in my calculations because the integration becomes somewhat much more difficult(gamma function and stuff like that). Are there any helpful inequalities that I'm missing or do I need alternate perspective in my calculations? Regards, Raxel.
Holder's inequality may be of some use here, although you only really need the special case of the Cauchy-Schwarz inequality. This says that if $f,g \in L^{2}$ then
$$\left| \int f(x)\bar{g}(x) dx \right|^{2} \leq \int |f(x)|^2 dx \cdot \int |g(x)|^2 dx$$