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Complete monotonicity of solutions of the Abel equation F(ex)=F(x)+1

Hilberdink Titus

Bulletin of the Polish Academy of Sciences. Mathematics

2023

Vol. 71, no 2

135--146

We investigate the functions F : R → R which are C ∞ solutions of the Abel functional equation F(ex) = F(x) + 1. In particular, we determine the asymptotic behaviour of the derivatives and show that no solution can have F′completely monotonic on any interval (α, ∞). We discuss what could be considered the best behaved solution of this equation.

HCM property and the half-Cauchy distribution

Bosch P.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 2

191--200

Let Zα and Zα be two independent positive α-stable random variables. It is known that (Zα/Zα)α is distributed as the positive branch of a Cauchy random variable with drift. We show that the density of the power transformation (Zα/Zα)β is hyperbolically completely monotone in the sense of Thorin and Bondesson if and only if α ≤ 1/2 and |β| ≥ α/(1−α). This clarifies a conjecture of Bondesson (1992) on positive stable densities.