Radius problems associated with pre-Schwarzian and Schwarzian derivatives

✍ Scribed by Ponnusamy, Saminathan; Sahoo, Swadesh Kumar; Sugawa, Toshiyuki

• Book ID: 125523840
• Publisher: Oldenbourg Wissenschaftsverlag
• Year: 2014
• Tongue: English
• Weight: 284 KB
• Volume: 34
• Category: Article
• ISSN: 0174-4747
• DOI: 10.1515/anly-2012-1219

✦ Synopsis

Some of important univalence criteria for a non-constant meromorphic function 𝑓(𝑧) on the unit disk 𝔻 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 ∈ (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.