  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
$\mathcal{F}[A,B].$
