Radius problems associated with pre-Schwarzian and Schwarzian derivatives

Details Radius problems associated with pre-Schwarzian and Schwarzian derivatives Radius problems associated with pre-Schwarzian and Schwarzian derivatives

2012-03-18

arxiv.org/abs/1203.3918v1

Mathematics

Complex Variables

8 pages, submitted to a journal

Scientific paper

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

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