Assume $\gamma $ is a $C^1$ closed curve in the complex plane whose length is $2\pi$,and consider all its possible regular parametrization throught a parameter $t \in [0,2\pi]$
Let $\gamma(t)$ one of these parametrizations and let $\gamma(t) = \sum_{n=-\infty}^{\infty} a_n e^{itn}$ its fourier series. Prove that
$\sum _{n= -\infty}^{\infty}|a_n|^2n^2 \geq \ 1$
and equality holds if and only if the parametrization is the one using arc length as parameter.
It is clear that I have to derive the fourier series and then use parseval identity but I do not know how to conclude. I would like a full solution to this problem.
By $||\cdot||$ I mean the $L^2[0,2\pi]$ norm.
$$\sum n^2|a_n|^2=\frac{1}{2\pi}\int_0^{2\pi}|\gamma'(t)|^2dt=\frac{1}{2\pi}||\gamma'||^2\ge^a\frac{1}{2\pi} \frac{\langle \gamma',1\rangle^2}{||1||^2}= \frac{1}{2\pi}\frac{1}{2\pi}\left(\int_0^{2\pi}|\gamma'(t)|dt\right)^2=\frac{1}{4\pi^2}l(\gamma)^2=1$$