On a Class of certain Non-Univalent Functions

Abstract: In this paper, we introduce a family of analytic functions given by $$\psi_{A,B}(z):= \dfrac{1}{A-B}\log{\dfrac{1+Az}{1+Bz}},$$ 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 $\mathcal{F}[A,B].$