4
$\begingroup$

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?

Improve this question
$\endgroup$
2
  • 5
    $\begingroup$ in general you can get bounds for the in-between derivatives from the outer ones (in other words bounds for $f, f^{(n)}$ give bounds for $f',..f^{(n-1)}, n \ge 2$; this comes under the Landau-Kolmogorov inequalities heading en.wikipedia.org/wiki/Landau%E2%80%93Kolmogorov_inequality $\endgroup$
    – Conrad
    Commented Apr 19, 2022 at 23:46
  • $\begingroup$ Thank you, Conrad! That's really interesting. $\endgroup$
    – Seven9
    Commented Apr 20, 2022 at 12:02

1 Answer 1

Reset to default
11
$\begingroup$

No, $f(x)=\int_0^x \sin t^2\, dt$ is a counterexample.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Thank you, Christian! $\endgroup$
    – Seven9
    Commented Apr 20, 2022 at 12:01

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .