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 S^*ₛ and S^*ρ, where S^*ₛ is the set of all normalized analytic functions f satisfying zf'(z)/f(z) 1+sin z and S^*ρ is the set of all normalized analytic functions f satisfying zf'(z)/f(z) 1+⁻¹ z.