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][1] (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][2] (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
$$\sum_{|\lambda|\le N}c(\lambda+1)=2P-1,\qquad \sum_{|\lambda|\le N}c(\lambda)=P.$$


  [1]: http://link.springer.com/article/10.1007%2FBF01087239
  [2]: http://iam.khv.ru/articles/Ustinov/nth01_eng.pdf