Criteria for Starlikeness Using Schwarzian Derivatives

TL;DR

This paper establishes new sufficient conditions for a normalized analytic function to be starlike in the unit disk, using criteria involving quotients related to derivatives and the Schwarzian derivative, based on differential subordination theory.

Contribution

It introduces novel criteria for starlikeness based on Schwarzian derivatives and differential subordination, expanding the theoretical framework for geometric function analysis.

Findings

Derived multiple sufficient conditions for starlikeness.

Connected starlikeness criteria with Schwarzian derivative properties.

Applied admissibility criteria from differential subordination theory.

Abstract

For a normalised analytic function f defined on the open unit disk in the complex plane, we determine several sufficient conditions for starlikeness in terms of the quotients Q_{ST}:=zf'(z)/f(z), Q_{CV}:=1+zf"(z)/f'(z) and the Schwarzian derivative Q_{SD}:=z^2((f"(z)/f'(z))'-(f"(z)/f'(z))^2/2)$. These conditions were obtained by using the admissibility criteria of starlikeness in the theory of second order differential subordination.