Decay of Fourier Transform

Question

I encountered the following statement, and I cannot see why it is true(if it is).

Suppose $f$ is a nonnegative, bounded, compactly supported and measurable function with the following properties: $||f||_1=1$, $|\widehat f(y)|<1$ for $y\neq 0$, $|\widehat f(0)|=1$ and $\frac{d}{dy}|\widehat f(y)|^2<0$.

Then, the claim which I don't quite see how it follows is the following:

For small $K$ and some $r>0$ we have $$\sup_{|y|\geq K}|\widehat f(y)|^2\leq e^{-r|K|^2}.$$

From the Riemann-Lebesgue lemma all we would get is decay of the order $1/K^2.$

Thanks

Answer 1

Issue Resolved! I will post a correct statement and an answer when I have some time in the next few days!