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 $$ \int_0^\infty f(x)\cos(x\omega) \, dx\ge 0. $$ The function $f(x)=\exp(-x^{\alpha})[x^{\alpha} \log(x) + \delta/\alpha ]$, $\alp...

Let $X$ be an absolutely continuous (i.e. its law is absolutely continuous with respect to the Lebesgue measure) random variable with probability density $p$. Its differential entropy is given by $$h(X) = - \int_{\mathbb{R}} p(x) \log p(x) \mathrm{d} x$$ with the convention $0 \log 0 = 0$, as soo...