Which functions have all derivatives everywhere positive? - MathOverflow

Which functions have all derivatives everywhere positive?

2012-01-26T17:41:17Z

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?

Answer by Stefan Waldmann for Which functions have all derivatives everywhere positive?

2012-01-26T17:51:14Z

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...

Answer by Gerald Edgar for Which functions have all derivatives everywhere positive?

2012-01-26T18:13:37Z

See completely monotonic in the literature. Function $f(x)$ is completely monotonic if and only if $f(-x)$ is one of yours.

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

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

Answer by Liviu Nicolaescu for Which functions have all derivatives everywhere positive?

2012-01-26T18:14:58Z

If $f(x)$ is a function with positive derivatives, then the function $f(-x)$ is completely monotonic. The completely monotonic functions are clasified by Bochner's theorem

Nimza (mathoverflow.net/users/17896), On the generalisation of Bernstein's theorem on monotone functions, http://mathoverflow.net/questions/86524 (version: 2012-01-24)