The function φL defined by φL(z)=√1+z maps the unit disk D onto Ω={w∈C:|w2−1|<1}, the region in the right half-plane bounded by the lemniscate of Bernoulli |w2−1|=1. This paper deals with starlike functions defined on D with zf′(z)/f(z)∈Ω or equivalently zf′(z)/f(z) is subordinated to φL(z) and these functions are related to the analytic function p:D→C with p(z)∈Ω for all z∈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)∈Ω. As applications, we give several sufficient conditions for functions f to satisfy zf′(z)/f(z)∈Ω.
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