A particular family of absolutely monotone functions and relations to branching processes. (English) Zbl 1538.26025

Möhle, M.

Anal. Math. 49, No. 2, 641-650 (2023).

A function \(f: [0,1)\to\mathbb{R}\) is absolutely monotone if \(f^{(n)}(z)\ge 0\) for all \(n\ge 0\) and \(z\in [0,1)\). Clearly, if \(f\) is absolutely monotone, then all its derivatives are absolutely monotone, too. It is known that the following functions \(f: [0,1)\to \mathbb{R}\) are absolutely monotone:
\(f(z)=-\ln(1-z)\);
\(f(z)=\frac{1}{1-z}\);
\(f(0)=0\) and \(f(z)=1+\frac{z}{\ln(1-z)}\) for \(z\in (0,1)\).
In the paper under review, the author completes this list by showing that the map \[f(z)=\ln\left( 1-\frac{\ln(1-z)}{c}\right)\] is absolutely monotone if and only if \(c\ge 1\). Consequently, all derivatives of \(f\) are absolutely monotone if \(c\ge 1\). The motivation for studying such functions comes from the theory of continuous-time branching processes.

Reviewer: Tomasz Natkaniec (Gdańsk)

MSC:

26A48	Monotonic functions, generalizations
33B30	Higher logarithm functions
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords:

absolute monotonicity; continuous-time branching process; Gautschi’s double inequality; integral representation; special functions; Stirling numbers

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