Condition for non-negative Fourier transform: $\mathcal{F} \exp(-|x|^a)(|x|^a\log(x)+\delta) \ge 0$

Question

I am looking for $\delta>0$, such that Fourier transform of $\exp(-|x|^a)(|x|^a\log(x)+\delta)$, $a \in (1,2)$ is non-negative. I have tried to evaluate the integral for $a=2$. I failed to arrive at an analytical expression to be of any use. I could prove that such $\delta$ exists, again for $a=2$ only