Radius problems associated with pre-Schwarzian and Schwarzian derivatives

S. Ponnusamy; S. K. Sahoo; T. Sugawa

Radius problems associated with pre-Schwarzian and Schwarzian derivatives

Complex Variables 2012-03-20 v1

Authors: S. Ponnusamy , S. K. Sahoo , T. Sugawa

Abstract

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

Keywords

pre-Schwarzian derivative Schwarzian derivative radius problem univalence criteria meromorphic function unit disk normalized univalent functions

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@article{arxiv.1203.3918,
  title  = {Radius problems associated with pre-Schwarzian and Schwarzian derivatives},
  author = {S. Ponnusamy and S. K. Sahoo and T. Sugawa},
  journal= {arXiv preprint arXiv:1203.3918},
  year   = {2012}
}

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8 pages, submitted to a journal

Radius problems associated with pre-Schwarzian and Schwarzian derivatives

Abstract

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