Higher order differential subordinations for certain starlike functions

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 class of all normalized analytic functions $f$ satisfying $ zf'(z)/f(z)\prec 1+\sin z$ and $\mathcal{S}^*_{\rho}$ is the class of all normalized analytic functions $f$ satisfying $ zf'(z)/f(z)\prec 1+\sinh^{-1} z$.