Inequality in fourier analysis lecture

This is problem

Suppose $f:\mathbb{T}\rightarrow \mathbb{C}$, and $f$ has continuous derivative on $\mathbb{T}$ then: $$\sum_{-\infty}^{\infty}|\hat{f}(n)|\leq |\hat{f}(0)|+\Big{(}\frac{\pi}{6}\int_\mathbb{T}|f'(t)^2|dt\Big{)}^{1/2}$$

using the Bessel's inequality, and cauchy schwarz inequaliy. and the fact $\sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi}{6}$

and $\mathbb{T}$ is torus here.

this is my approach:

from the $C^1$ condition, for $n\in\mathbb{Z}-\{0\}$ $$\hat{f}(n)=\frac{1}{2\pi}\int_{\mathbb{T}}f(t)\exp(-int)dt=\frac{1}{2\pi in}\int_\mathbb{T}f'(t)\exp(-int)dt$$

by integration by part.

here I don't know how we can use the Bessel's inequality and Cauchy schwarz inequality. please give me some help.

$$\sum_{n\ne 0}|\hat{f}(n)| \le \sqrt{\sum_{n\ne 0} n^2 |\hat{f}(n)|^2} \sqrt{\sum_{n\ne 0} \frac1{n^2}}$$ Then apply the Bessel inequality to $\sum_{n\ne 0} n^2 |\hat{f}(n)|^2$