TITLE: Growth of a function with every derivative everywhere increasing
QUESTION [6 upvotes]: If a function $f: \mathbb{R} \to \mathbb{R}^{+}$, is infinitely differentiable everywhere, and is such that every derivative is monotonically strictly increasing everywhere, then what is the slowest-growing function that $f$ could be? What if the derivatives are just monotonically nondecreasing everywhere?
Feel free to add in additional properties (which you would specify) if this question is not well-formulated enough.
This question just popped into my head. I have suspicions and will work on it too.

REPLY [2 votes]: The OP property means that $f^{(k)}(x) \ge 0, k \ge 2$ (or strict inequality if needed); this means $f''$ is absolutely monotonic on $\mathbb R$ and by Bernstein's theorem it is (entire) and of the form $f(x)=\int_0^{\infty}e^{xt}d\mu(t)$ where $\mu$ is a positive measure on $[0,\infty)$ for which the integral is absolutely convergent for all $x \in \mathbb C$ (which is actually equivalent to absolute convergence for all $x >R$).
There is a trivial case when $f''=a \ge 0$ (in the $\ge$ case only of course) corresponding to $f(x)$ a quadratic which is positive on $\mathbb R$, but otherwise, the support of the measure cannot be concentrated at zero only so there is $\mu[\alpha, \beta] >0, \alpha >0$ and trivially $f''(x) > \mu [\alpha, \beta]e^{\alpha x}, x >0$ hence $f''$ is an entire function of at least exponenttial growth and so is $f$; by taking $f(x)=e^{\epsilon x}, \epsilon >0$ we obviously do not have a "lowest growth" as such
Note that the non-trivial part is absolute monotonicity to the left (ie on $(-\infty, a)$ as for $(a, \infty), a$ finite anything of the form $\sum {a_n(x-a)^n}$ with $a_n > 0$ radius of convergence infinite will do, so Bernstein theorem is stated that way usually - ie absolute monotonicity on the negative axis implies the integral representation and the continuation to the positive axis follows by analytic continuation - sometimes the results are stated also in terms of complete monotonicity - derivatives alternate in sign - on the positive axis or to the right more generally, and the results are equivalent with $x \to -x$)