On a Class of certain Non-Univalent Functions

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