I am trying to find a "manageable" sufficient condition for a function $f\colon \mathbb{R} \to \mathbb{R}$ to satisfy $$ |f^{(n)}(x)| \le \frac{Mn!}{(x-x_0)^{n+1}}\tag{1} $$ for some $M \ge 0$, $x_0 \in \mathbb{R}$ and all $n \ge 0$, $x > x_0$.
By Widder's theorem, it follows that for $x_0 = 0$ the above condition is equivalent to the fact that the inverse Laplace transform of $f$ is a bounded function. However, this is rather difficult to check for a nontrivial $f$.
Is there any sufficient condition that guarantees $(1)$ and is possible to check?
For example, I would like to show that the function $$ f(x) = \frac{x \cosh x}{(x^2+1) \sinh x + 2x \cosh x} $$ has the property.