Let $N$ be nonnegative integer, $F(x)$ be a nonnegative real Lebesgue integrable function defined on $[0,1]$. Suppose that all Fourier coefficients $c(\lambda)=\int_0^1F(x)e^{-2\pi i \lambda x}dx$ of the function $F(x)$ are nonnegative reals. In the article System of inequalities (1984) Bykovskii proved that for any integer $\mu$ $$\sum_{|\lambda|\le N}c(\lambda+\mu)\le 4\sum_{|\lambda|\le N}c(\lambda).$$
Is the constant $4$ best possible?
Similar proof can be found here (see Lemma 2).
EDT. From fedja's comment follows that optimal constant is not larger than $3$. Also it is not less than $2$ because we can take the function $$F(x)=\left|\sum_{k=1}^Pe^{2\pi i (N+1)kx}\right|^2.$$ For this function $c(0)=P$, $c(N+1)=P-1$, so $$\sum_{|\lambda|\le N}c(\lambda+1)=2P-1,\qquad \sum_{|\lambda|\le N}c(\lambda)=P.$$