Strict definiteness of integrals via complete monotonicity of derivatives

Let $k$ be a nonnegative integer and let $\varphi : (0,\infty ) \rightarrow \Bbb R$ be a $C^\infty$ function with $(-)^k\cdot \varphi ^{(k)}$ completely monotone and not constant. If $\sigma \neq 0$ is a signed measure on any euclidean space $\Bbb R^d$, with vanishing moments up to order $k-1$, then the integral $\int _{\Bbb R^d} \int _{\Bbb R^d} \varphi ( \|x-y\|^2 ) \, d\sigma (x) d\sigma (y)$ is strictly positive whenever it exists. For general $d$ no larger class of continuous functions $\varphi$ seems to admit the same conclusion. Examples and applications are indicated. A section on ''bilinear integrability'' might be of independent interest.