Inclusion Criteria for Subclasses of Functions and Gronwall's Inequality

TL;DR

This paper establishes new criteria involving Schwarzian derivatives and second coefficients to determine when normalized analytic functions are univalent or belong to specific subclasses, including strongly α-Bazilevič functions.

Contribution

It introduces novel conditions based on Schwarzian derivatives and second coefficients for classifying analytic functions into various subclasses.

Findings

Derived sufficient conditions for univalence in the unit disk.

Established criteria for functions to be strongly α-Bazilevič of order β.

Linked second coefficient bounds to function subclass membership.

Abstract

A normalized analytic function f is shown to be univalent in the open unit disk D if its second coefficient is sufficiently small and relates to its Schwarzian derivative through a certain inequality. New criteria for analytic functions to be in certain subclasses of functions are established in terms of the Schwarzian derivatives and the second coefficients. These include obtaining a sufficient condition for functions to be strongly $\alpha$-Bazilevi\^c of order $\beta$.