How can we find functions on $\mathbb{R}$ with exponential-like properties, namely:
- $f(x)$ is infinitely differentiable;
- $f(x)$ and all its derivatives are monotonic;
- $f(x)$ and all its derivatives obey the following limits:
$$\lim_{x \to -\infty}f(x)=0$$
$$\lim_{x \to +\infty}f(x)=+\infty$$
One such function is obviously the exponent itself ($a,b$ - real positive constants):
$$f(x)=ae^{bx}$$
Another function which seems to have these properties (I don't know how to prove it) is the 'Sophomore's function':
$$s(x)=\int_0^1 u^{-u~x} du=\sum_{k=1}^{\infty} \frac{x^{k-1}}{k^k}$$
For the proof of the integral formula see this answer by Sangchul Lee.
The derivatives are easy to find (both for the series and the integral formula) and they all seem to obey the above properties:
How can we find other such functions?
And (related) how to prove that $s(x)$ has these properties?
If $g(x) \ge 0$, $\lim_{x \to -\infty} g(x) = 0$, and $\int_{-\infty}^{\infty} g(t) dt = \infty $, then $f(x) =\int_{-\infty}^x g(t) dt $ is such a function.
(added a bit later)
If you want all the derivatives to be monotonic, impose that restriction on $g$.