Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a bounded analytic function such that its derivative is also bounded. What kind of bound can we get on the higher order derivatives of $f$? Does it follow that they are bounded as well?



Bounded real analytic function with bounded derivative and its higher order derivatives
No, $f(x)=\int\_0^x \sin t^2\, dt$ is a counterexample.


