  For an analytic function $f$ on the unit disk $\mathbb{D}=\{z:|z|<1\}$
satisfying $f(0)=0=f'(0)-1,$ we obtain sufficient conditions so that $f$
satisfies $|(zf'(z)/f(z))^2-1|<1.$ The technique of differential subordination
of first or second order is used. The admissibility conditions for lemniscate
of Bernoulli are derived and employed in order to prove the main results.
