TITLE: Which functions have all derivatives everywhere positive?
QUESTION [16 upvotes]: 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?

REPLY [20 votes]: 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

REPLY [14 votes]: Well, there are certainly more. If you look at the chain rule then you see that the $n$-th derivative is a linear combinations of products of derivatives of the two functions you compose with positive coefficients. Thus if you have two functions with your property, then their composition will again have only positive derivatives. So you can go on...

REPLY [8 votes]: If $f(x)$ is a function with positive  derivatives, then the function $f(-x)$ is completely monotonic. The completely monotonic functions are classified by Bochner's theorem; see Nimza's question On the generalisation of Bernstein's theorem on monotone functions.