Some Problems Related to Completely Monotone Positive Definite Functions

This paper deals with several problems related to functions of the class ${\mathcal{CM}}$ of completely monotone functions and functions of the class $\Phi(E)$ of positive definite functions on a real linear space $E$. Theorem 1 verifies some conjectures of Moak related to the complete monotonicity of the function $x^{-\mu}(x^2+1)^{-\nu}$. Theorem 2 states that if $f\in C^{\infty}{(0,+\infty)}$ and $\delta\in{\mathbb{R}}$, then
$$ f(x)-a^\delta f(a x)\in {\mathcal{CM}}\qquad \text{for all}\quad a>1 $$
if and only if $-\delta f(x)-xf'(x)\in \mathcal{CM}$. A similar result for functions in $\Phi(E)$ is obtained in Theorem 9: if $\varepsilon\in{\mathbb{R}}$ and a function $h\colon [0,+\infty)\to\mathbb{R}$ is continuous on $[0,+\infty)$ and differentiable on the interval $(0,+\infty)$ and satisfies the condition $xh'(x)\to 0$ as ${x\to+0}$, then
$$ h(\rho(u))-a^{-\varepsilon}h(a\rho(u))\in\Phi(E)\qquad \text{for all}\quad a>1 $$
if and only if $ \psi_{\varepsilon}(\rho(u))\in\Phi(E), $ where $\psi_{\varepsilon}(x):=\varepsilon h(x)- xh'(x)$ for $x>0$ and $\psi_{\varepsilon}(0):=\varepsilon h(0)$. Here $\rho$ is a nonnegative homogeneous function on $E$ and $\rho(u)\not\equiv 0$. It is proved (Example 6) that:

• $e^{-\alpha\|u\|}(1-\beta\|u\|)\in\Phi(\mathbb{R}^m)$ if and only if $-\alpha\le\beta\le\alpha/m$;
• $e^{-\alpha\|u\|^2}(1-\beta\|u\|^2)\in\Phi({\mathbb{R}}^m)$ if and only if $0\le\beta\le2\alpha/m$.

Here $\|u\|$ is the Euclidean norm on $\mathbb{R}^m$. Theorem 11 deals with the case of radial positive definite functions $h_{\mu,\nu}$.