Starlikeness of Certain Non-Univalent Functions

TL;DR

This paper investigates the starlikeness properties of certain non-univalent functions defined via classes of analytic functions with positive real part, determining radii for these functions to belong to various starlike subclasses.

Contribution

It introduces new classes of non-univalent functions and establishes radius results for their inclusion in multiple starlike subclasses, expanding the understanding of geometric properties of these functions.

Findings

Determined radii for functions to be starlike of order α

Established radii for parabolic and lemniscate starlike functions

Analyzed functions related to sine, cardioid, lune, nephroid, and sigmoid shapes

Abstract

We consider three classes of functions defined using the class $\mathcal{P}$ of all analytic functions $p(z)=1+cz+\dotsb$ on the open unit disk having positive real part and study several radius problems for these classes. The first class consists of all normalized analytic functions $f$ with $f/g\in\mathcal{P}$ and $g/(zp)\in\mathcal{P}$ for some normalized analytic function $g$ and $p\in \mathcal{P}$. The second class is defined by replacing the condition $f/g\in\mathcal{P}$ by $|(f/g)-1|<1$ while the other class consists of normalized analytic functions $f$ with $f/(zp)\in\mathcal{P}$ for some $p\in \mathcal{P}$. We have determined radii so that the functions in these classes to belong to various subclasses of starlike functions. These subclasses includes the classes of starlike functions of order $\alpha$, parabolic starlike functions, as well as the classes of starlike functions associated with lemniscate of Bernoulli, reverse lemniscate, sine function, a rational function, cardioid, lune, nephroid and modified sigmoid function.