Abstract:
This paper deals with several problems related to functions of the class CM 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(x)−aδf(ax)∈CMfor alla>1 if and only if −δf(x)−xf′(x)∈CM. 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,+∞) and differentiable on the interval (0,+∞) and satisfies the condition xh′(x)→0 as x→+0, then h(ρ(u))−a−εh(aρ(u))∈Φ(E)for alla>1 if and only if ψε(ρ(u))∈Φ(E), where ψε(x):=εh(x)−xh′(x) for x>0 and ψε(0):=εh(0). Here ρ is a nonnegative homogeneous function on E and ρ(u)≢0. It is proved (Example 6) that:
e−α‖u‖(1−β‖u‖)∈Φ(Rm) if and only if −α⩽β⩽α/m;
e−α‖u‖2(1−β‖u‖2)∈Φ(Rm) if and only if 0⩽β⩽2α/m.
Here ‖u‖ is the Euclidean norm on Rm. Theorem 11 deals with the case of radial positive definite functions hμ,ν.
This publication is cited in the following 2 articles:
Henry J. Brown, Yury Grabovsky, “On Feasibility of Extrapolation of Completely Monotone Functions”, SIAM J. Math. Anal., 56:6 (2024), 7713
V. Zastavnyi, A. Manov, “Some generalizations of the problem of positive definiteness of a piecewise linear function”, Journal of Mathematical Analysis and Applications, 519:2 (2023), 126864