  In this paper, we employ a novel second and third-order differential
subordination technique to establish the sufficient conditions for functions to
belong to the classes $\mathcal{S}^*_s$ and $\mathcal{S}^*_{\rho}$, where
$\mathcal{S}^*_s$ is the set of all normalized analytic functions $f$
satisfying $ zf'(z)/f(z)\prec 1+\sin z$ and $\mathcal{S}^*_{\rho}$ is the set
of all normalized analytic functions $f$ satisfying $ zf'(z)/f(z)\prec
1+\sinh^{-1} z$.
