Let $f\in \mathcal L^1(\mathbb R^\nu)$, $f>0$. I want to show that $$|\hat f(\xi)|<\hat f(0) \qquad \forall \xi \neq 0$$ It is easy to prove that $$|\hat f(\xi)|\le\hat f(0) $$ In fact $$|\hat f(\xi)|=\left |\int_\mathbb R e^{-i\langle x,\xi\rangle }f(x)\mathrm dx \right |\le \int_ \mathbb R |e^{-i\langle x,\xi\rangle}||f(x)|\mathrm dx=\int_\mathbb R f(x)\mathrm d x=\hat f(0)$$ but I am having a hard time proving that such maximum cannot be achieved by a non-zero $\xi$.
I tried to suppose by contradiction that there exists $\bar\xi\neq 0$ such that $$|\hat f(\bar \xi)|=\hat f(0)$$ but I was not able to show a contradiction. There probably is a one liner answer lying somewhere.
Question: how can we prove that such $\bar \xi$ cannot exist?