On a Class of certain Non-Univalent Functions

TL;DR

This paper introduces a new class of analytic functions defined via subordination to specific mappings, analyzing their properties, and determining sharp radii for starlikeness and univalence, with applications to related function classes.

Contribution

It defines the class [A,B] using a novel subordination approach and establishes sharp radii for starlikeness and univalence, extending the understanding of non-univalent functions.

Findings

Sharp radius of starlikeness of order elta determined.

Sharp radii for univalence established.

Characterization of [A,B] properties and applications to related classes.

Abstract

In this paper, we introduce a family of analytic functions given by $$\psi_{A,B}(z):= \dfrac{1}{A-B}\log{\dfrac{1+Az}{1+Bz}},$$ which maps univalently the unit disk onto either elliptical or strip domains, where either $A=-B=\alpha$ or $A=\alpha e^{i\gamma}$ and $B=\alpha e^{-i\gamma}$ ($\alpha\in(0,1]$ and $\gamma\in(0,\pi/2]$). We study a class of non-univalent analytic functions defined by \begin{equation*} \mathcal{F}[A,B]:=\left\{f\in\mathcal{A}:\left( \dfrac{zf'(z)}{f(z)}-1\right)\prec\psi_{A,B}(z)\right \}. \end{equation*} Further, we investigate various characteristic properties of $\psi_{A,B}(z)$ as well as functions in the class $\mathcal{F}[A,B]$ and obtain the sharp radius of starlikeness of order $\delta$ and univalence for the functions in $\mathcal{F}[A,B]$. Also, we find the sharp radii for functions in $\mathcal{BS}(\alpha):=\{f\in\mathcal{A}:zf'(z)/f(z)-1\prec z/(1-\alpha z^2),\;\alpha\in(0,1)\}$, $\mathcal{S}_{cs}(\alpha):=\{f\in\mathcal{A}:zf'(z)/f(z)-1\prec z/((1-z)(1+\alpha z)),\;\alpha\in(0,1)\}$ and others to be in the class $\mathcal{F}[A,B].$