Let $p(x)=\sum_{k=1}^m [a_k\cos(n_kx)+b_k\sin(n_kx)]$ be a null average trigonometric polynomial (null average means that is $\int_\mathbb T p =0$ or, equivalently, there are no $a_0$ and $b_0$). Denote with $N$ the order of $p$, namely the number of times that $(a_k,b_k)\neq(0,0)$. We assume that $\int_\mathbb T p^2=1$.

I have done some numerical computations and I think the following may be true, there is an optimal constant $C(N)>0$ depending only on $N$, such that

$$\frac12\|p\|_{L^1(\mathbb T)}=\int\limits_{\{p>0\}\cap\mathbb T} p(s)\,ds=-\int\limits_{\{p<0\}\cap\mathbb T} p(s)\,ds\geq C(N).$$

The optimal $C(N)$ actually exists, but I am interested in an explicit expression or a way to find $C(N)$. When $N=2$ clearly the equality holds and $C(2)=\frac{2}{\sqrt{\pi}}$. Thank you.