Some Problems Related to Completely Monotone Positive Definite Functions

Abstract

This paper deals with several problems related to functions of the class \({\mathcal C}{\mathcal M}\) of completely monotone functions and functions of the class Φ(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^−μ (x²+ 1)^−ν. Theorem 2 states that if f ∈ C^∞(0, + ∞) and δ ∈ ℝ, then

\(f\left( x \right) - {a^\delta }f\left( {ax} \right)\; \in \;{\mathcal C}{\mathcal M}\;\;\;\;\;\;{\rm{for}}\;{\rm{all}}\;\;\;\;a > 1\)

if and only if \( - \delta f\left( x \right) - xf\prime \left( x \right)\; \in \;{\cal C}{\cal M}\). A similar result for functions in Φ(E) is obtained in Theorem 9: if ε ∈ ℝ and a function h:[0, + ∞) → ℝ is continuous on [0, +œ) and differentiable on the interval (0, + œ) and satisfies the condition xh′ (x) → 0 as x → +0, then

\(h\left( {\rho \left( u \right)} \right) - {a^{ - \varepsilon }}h\left( {a\rho \left( u \right)} \right)\; \in \;{\rm{\Phi }}\left( E \right)\;\;\;\;\;\;{\rm{for}}\;{\rm{all}}\;\;\;\;a > 1\)

if and only if ψ_ε(p(u)) ∈ Φ(E), where ip_ε(x):= εh(x) − xh^′(x) for x > 0 and ψ_ε(0): = εh(0). Here p is a nonnegative homogeneous function on E and p(u) ≢ 0. It is proved (Example 6) that: (1) e^−α∥u∥ (1 − β∥u∥) ∈ Φ(ℝ^m) if and only if −α ≤ β ≤ a/m;(2) e^{−α∥u∥2} (1 − β∥u∥²) ∈ Φ(ℝ^m) if and only if 0 ≤ β ≤ 2α/m. Here ∥u∥ is the Euclidean norm on ℝ^m. Theorem 11 deals with the case of radial positive definite functions h_μ,ν.