Let $a_n$ and $b_n$ the fourier coefficient of a $2\pi-$ periodic functions $f$ we assume the regularity as we want to obtain the convergence of the series ( for example $f$ is $C^2$)
How can i prove the inequality (if it's true) $$\sum_{k=2}^{\infty} (a_{k}^{2}+b_{k}^{2}) \le C \sum_{k=2}^{\infty}(k^{2}-1) (a_{k}^{2}+b_{k}^{2}) $$
and what is the best constant $C\in \mathbb{R}_{+}$