On a Class of certain Non-Univalent Functions

In this paper, we introduce a family of analytic functions given by $ψ_{A,B}(z):= \dfrac{1}{A-B}\log{\dfrac{1+Az}{1+Bz}},$ which maps univalently the unit disk onto either elliptical or strip domains, where either $A=-B=α$ or $A=αe^{iγ}$ and $B=αe^{-iγ}$ ($α\in(0,1]$ and $γ\in(0,π/2]$). 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 $ψ_{A,B}(z)$ as well as functions in the class $\mathcal{F}[A,B]$ and obtain the sharp radius of starlikeness of order $δ$ and univalence for the functions in $\mathcal{F}[A,B]$. Also, we find the sharp radii for functions in $\mathcal{BS}(α):=\{f\in\mathcal{A}:zf'(z)/f(z)-1\prec z/(1-αz^2),\;α\in(0,1)\}$, $\mathcal{S}_{cs}(α):=\{f\in\mathcal{A}:zf'(z)/f(z)-1\prec z/((1-z)(1+αz)),\;α\in(0,1)\}$ and others to be in the class $\mathcal{F}[A,B].$

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