Some remarks on certain Bazilevic functions

TL;DR

This paper explores conditions under which analytic functions belong to Bazilevic subclasses and examines their closure properties under Bernardi integrals, contributing to the understanding of function class behaviors.

Contribution

It provides new sufficient conditions for functions to be in Bazilevic subclasses and investigates their closure properties under Bernardi integrals.

Findings

Established new sufficient conditions for Bazilevic functions.

Analyzed the closure properties of these classes under Bernardi integrals.

Extended previous results by considering the converse problem.

Abstract

In this note we give some sufficient conditions for an analytic function $f(z)$ normalized by $f'(0)=1$ to belong to certain subfamilies of the class of Bazilevic functions. In earlier works, the closure property of many classes of functions under the Bernardi integral have been considered. The converse of this problem is also considered here.