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

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.

MSC:

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

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