TITLE: Looking for sufficient conditions for positive Fourier transforms
QUESTION [12 upvotes]: I am looking for some sufficient conditions for an even, continuous, nonnegative, non-increasing, non-convex function to be non-negative definite.  In other words
$$
\int_0^\infty f(x)\cos(x\omega) \, dx\ge 0, \quad   \omega \in \mathbb{R}.
$$
The function $f(x)=\exp(-x^{\alpha})[x^{\alpha} \log(x) + \delta/\alpha ]$, $\alpha\in(1,2)$, $\delta\ge 2$.  Note that for $\alpha\in(0,1)$, $f(x)$ is convex.
I have tried complex detour numerically (as in Tuck, E. O.: On positivity of Fourier transforms, Bull. Austral. Math. Soc., 74 (2006) 133–
138) – did not work out. It seems the imaginary part always increases around $0$.
I have tried Polya type conditions () - no luck either.
For $\alpha\in(0,1)$ everything works out as the function is convex.
Thank you in advance for any hints or references!

REPLY [2 votes]: How about the following reference T. Gneiting, Criteria of Po´lya type for radial positive definite functions, Proc. Am. Math. Soc. 129 (2001), 2309–2318. I have plotted the results for 6th derivative all numeric but it seems working,