On a Class of Certain Non-univalent Functions

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