Analytic function that map the unit disk into the inside of the lemniscate of Bernoulli

The function \begin{document}$ \varphi_L $\end{document} defined by id="M6">\begin{document}$ \varphi_L(z) = \sqrt{1+z} maps the unit disk id="M7">\begin{document}$ \mathbb{D} onto id="M8">\begin{document}$ \Omega \{w\in\mathbb{C}: |w^2-1|<1\} $\end{document}, region in right half-plane bounded lemniscate of Bernoulli id="M9">\begin{document}$ |w^2-1| 1 $\end{document}. This paper deals with starlike functions on id="M10">\begin{document}$ id="M11">\begin{document}$ zf'(z)/f(z)\in or equivalently id="M12">\begin{document}$ zf'(z)/f(z) is subordinated to id="M13">\begin{document}$ and these are related analytic id="M14">\begin{document}$ p:\mathbb{D}\to \mathbb{C} id="M15">\begin{document}$ p(z)\in for all id="M16">\begin{document}$ z\in id="M17">\begin{document}$ p(z) Using admissibility criteria first second order differential subordination, we investigate several subordination results id="M18">\begin{document}$ p satisfy id="M19">\begin{document}$ As applications, give sufficient conditions id="M20">\begin{document}$ f id="M21">\begin{document}$