I am looking for some sufficient conditions for an even, continuous, nonnegative, non increasing function $f(x)$ on $R$ such that 


$$
\int\_0^\infty \cos(xz) f(z) d z \ge 0 \qquad\text{for all $x\ge 0$.}
\tag{1}
$$


I have a such function $f$. It has a complicated form involving some special functions. But it is an even, continuous, nonnegative and non increasing function. The goal is to see if (1) is satisfied. A direct calculation is hard. 


Examples for these functions include: $(1+x^2)^{-1}$, $(1+|x|)^{-1}$, $\exp(-|x|)$, etc. One counterexample is $(1+x^4)^{-1}$.


This question is related the characterization of the nonnegative definite functions on $R$.


Thanks for any hints or references! 



A calculus question related to the nonnegative definite functions
see [On positivity of Fourier transforms](http://journals.cambridge.org/download.php?file=%2FBAZ%2FBAZ74_01%2FS0004972700047511a.pdf&code=3b5549fe1a8edfaf28e851f47f408cbf); one sufficient condition is that $f''(x)>0$ for all $x>0$.


there are other sufficient conditions, see for example [On the positivity of Fourier transforms](http://arxiv.org/abs/1405.3155)


a necessary and sufficient condition is not known.


