Welcome to MathOverflow

Question

3

1

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?

flag	asked Jan 26 2012 at 17:41 Will Sawin 18.5k●1●24●48

4

what happens for exp(exp(x))? – Yemon Choi Jan 26 2012 at 17:51

Gerald Edgar · Accepted Answer · 2012-01-26 18:13:37Z UTC

7

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

answered Jan 26 2012 at 18:13

Gerald Edgar
15.1k●1●28●67

3

You have to be careful, though, because in the literature completely monotonic functions are assumed to be defined only on the half-line, not on the whole line as in the question. – Mark Meckes Jan 26 2012 at 18:49

Yet we can reduce to the notion of completely monotonic functions on the positive half-line because $f:\mathbb{R}\to\mathbb{R}$ has all derivatives non-negative iff $f(c-x):[0,\infty)\to\mathbb{R}$ is completely monotonic for all $c\in\mathbb{R}$; this e.g. allows to exploit Bernstein's representation in the OP's case. – Pietro Majer Jan 26 2012 at 19:28

So we would say that for all $c$ there is a finite measure $\mu_c$ such that for $t\leq c$, $f(t)=\int_0^\infty e^{(t-c)x} d\mu$? We would then make sure that this implies that there is a measure $\mu$ such that for all $t$, $f(t)=\int_0^\infty e^{tx} d\mu$. – Will Sawin Jan 26 2012 at 19:45

Stefan Waldmann · Answer 2 · 2012-01-26 17:51:14Z UTC

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...

Liviu Nicolaescu · Answer 3 · 2012-01-26 18:14:58Z UTC

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)

Which functions have all derivatives everywhere positive?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

3 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.