  Some of important univalence criteria for a non-constant meromorphic function
$f(z)$ on the unit disk $\ID$ involve its pre-Schwarzian or Schwarzian
derivative. We consider an appropriate norm for the pre-Schwarzian derivative,
and discuss the problem of finding the largest possible $r\in (0,1)$ for which
the pre-Schwarzian norm of the dilation $r^{-1}f(rz)$ is not greater than a
prescribed number for normalized univalent functions $f(z)$ in the unit disk.
Similar results concerning the Schwarzian derivative are also obtained
