# Function has nth derivatives bounded by exponential

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.