Suppose $f,g$ are $L^2$ functions over $\mathbb R$ such that $\|f\|^2\ge \|g\|^2$ and for any $y\in \mathbb R$, $$ \int_{\mathbb R} f(x) \overline{g(x-y)} dx = 0.$$ If $F,G$ are their respective Fourier transform, is it true that $$|F(z)|\ge |G(z)| \quad \forall z?$$
This is the continuous version of majorization of polynomials over the unit circle. I thought it may give some insight to put it in another perspective, since I can't seem to be able to prove any of them.