Second and Third order differential subordination for exponential function

This article presents several findings regarding second and third-order differential subordination of the form: $$p(z)+{\gamma }_{1}z{p}^{\mathrm{\prime }}(z)+{\gamma }_2{z}^2{p}^{\mathrm{\prime }\mathrm{\prime }}(z)\prec h(z)⟹p(z)\prec e^z$$ and $$p(z)+{\gamma }_{1}z{p}^{\mathrm{\prime }}(z)+{\gamma }_2{z}^2{p}^{\mathrm{\prime }\mathrm{\prime }}(z)+{\gamma }_3{z}^3{p}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(z)\prec h(z)⟹p(z)\prec e^z\mathrm{.}$$ Here, ${\gamma }_1$, ${\gamma }_2$, and ${\gamma }_3$ represent positive real numbers, and various selections of $h(z)$ are explored within the context of the class ${\mathcal{S}}_e^{\ast }:=\{f\in \mathcal{A}:z{f}^{\mathrm{\prime }}(z)\mathrm{/}f(z)\prec e^z\}$, which denotes the class of starlike functions associated with the exponential function.