Assume

1. $F$ and  $f$ are non-negative  $2\pi$ periodic function that do not vanish identically.
2. $F$ is integrable on  $S_{1}$ and satisfies the orthogonality conditions
   $${\int}_{S_{1}}F(\theta)\cos\theta d\theta=0={\int}_{S_{1}}F(\theta)\sin\theta d\theta.$$
3. $f\in H^{1}(S_{1}).$

Then
$${\int}_{S_{1}}F(\theta) d\theta{\int}_{S_{1}}f(\theta) d\theta\geq 2\pi{\int}_{S_{1}}F(\theta)f(\theta) d\theta.$$