Consider the class of functions from $\mathbb R$ to $\mathbb R$, such that the function is positive everywhere and its $n$th derivative is positive everywhere for all $n$.


The only examples I can construct are the functions $ae^{bx}+c$ for $a,b,c>0$.


Are these functions the only examples?


If not, for which nonlinear functions $g$ does $e^{g(x)}$ have this property?



Which functions have all derivatives everywhere positive?
See **completely monotonic** in the literature. Function $f(x)$ is completely monotonic if and only if $f(-x)$ is the sort of function you're looking for.


S.N. Bernstein (1928). "Sur les fonctions absolument monotones". Acta Mathematica 52: 1–66. doi:10.1007/BF02592679.


<http://mathworld.wolfram.com/CompletelyMonotonicFunction.html>


