On a Class of certain Non-Univalent Functions

Abstract

In this paper, we introduce a family of analytic functions given by $${\psi }_{A,B}(z):=\frac{1}{A-B}\log \frac{1+Az}{1+Bz},$$ 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)\precψ_{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 $\mathcal{BS}(\alpha ):=\{f\in \mathcal{A}:z{f}^{\prime }(z)/f(z)-1\prec z/(1-\alpha z^2),\alpha \in (0,1)\}$, ${\mathcal{S}}_{cs}(\alpha ):=\{f\in \mathcal{A}:z{f}^{\prime }(z)/f(z)-1\prec z/((1-z)(1+\alpha z)),\alpha \in (0,1)\}$ and others to be in the class $\mathcal{F}[A,B].$