Does there exist an infinitely differentiable function $f: (0,\infty) \to \mathbb{R}$, other than a constant multiple of $e^{-x}$, satisfying $|f^{(n)}(x)| \leq e^{-x}$ for all $n$, $x$?
Some counterexamples we have excluded are $e^{-kx}$ for $k \not = 1$ (if $k < 1$ then the inequality fails for large $x$, if $k > 1$ then the inequality fails for large $n$) and the function $\frac{1}{1+e^{x}}$ (the higher derivatives blow up around $x=0$). We have tried using the Laplace transform to write $f(x) = \int_0^{\infty} g(t)e^{-tx}dt$, which shows we should expect higher derivatives to blow up if $f$ is not $e^{-x}$ but since $g$ need not be nonnegative we can't get anything concrete this way.