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).