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

The function $ \varphi_L $ defined by $ \varphi_L(z) = \sqrt{1+z} $ maps the unit disk $ \mathbb{D} $ onto $ \Omega = \{w\in\mathbb{C}: |w^2-1|<1\} $, the region in the right half-plane bounded by the lemniscate of Bernoulli $ |w^2-1| = 1 $. This paper deals with starlike functions defined on $ \mathbb{D} $ with $ zf'(z)/f(z)\in \Omega $ or equivalently $ zf'(z)/f(z) $ is subordinated to $ \varphi_L(z) $ and these functions are related to the analytic function $ p:\mathbb{D}\to \mathbb{C} $ with $ p(z)\in \Omega $ for all $ z\in \mathbb{D} $ by $ p(z) = zf'(z)/f(z) $. Using the admissibility criteria of the first and second order differential subordination, we investigate several subordination results for functions $ p $ to satisfy $ p(z)\in \Omega $. As applications, we give several sufficient conditions for functions $ f $ to satisfy $ zf'(z)/f(z)\in \Omega $.