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

Abstract

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