Radius problems associated with pre-Schwarzian and Schwarzian derivatives

TL;DR

This paper investigates the maximal radius within the unit disk where the pre-Schwarzian and Schwarzian derivatives of normalized univalent functions stay within certain bounds, enhancing understanding of univalence criteria.

Contribution

It introduces a norm-based approach to determine the largest radius for which univalence criteria involving derivatives are satisfied, extending previous univalence conditions.

Findings

Derived bounds for the radius related to pre-Schwarzian derivatives.

Established analogous results for Schwarzian derivatives.

Provided new insights into univalence criteria via derivative norms.

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