Higher order differential subordinations for certain starlike functions

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 ${\mathcal{S}}_s^{\ast }$ and ${\mathcal{S}}_{\rho }^{\ast }$, where ${\mathcal{S}}_s^{\ast }$ is the set of all normalized analytic functions $f$ satisfying $z{f}^{\mathrm{\prime }}(z)\mathrm{/}f(z)\prec 1+\sin z$ and ${\mathcal{S}}_{\rho }^{\ast }$ is the set of all normalized analytic functions $f$ satisfying $z{f}^{\mathrm{\prime }}(z)\mathrm{/}f(z)\prec 1+{\sinh }^{-1}z$.