# Totally monotonic

A function is said to be totally monotonic at $x_0$ if it has derivatives of any order and $$(-1)^{k+1} f^{(k)}(x_0) > 0$$ for every $k>0$.
Denote by $\bf{M}$ the class of real functions $F(x)$ defined for $x>0$ and totally monotonic at every $x>0$; $\log(x)$ is a typical representative of $\bf{M}$.
Theorem 2. Let $A(x)$ denote the principal Abel function of $e^x-1$. Then $A(x)\in \bf{M}$ and $A(x)$ is the only Abel function of $e^x-1$ with this property.