Bulletin of the
Korean Mathematical Society BKMS

This article presents several findings regarding second and third-order differential subordination of the form:
$$
p(z)+\gamma_1 zp'(z)+\gamma_2 z^2p''(z)\prec h(z)\implies p(z)\prec e^z
$$
and $$
p(z)+\gamma_1 zp'(z)+\gamma_2 z^2p''(z)+\gamma_3 z^3p'''(z)\prec h(z)\implies p(z)\prec e^z.
$$
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 $\mathcal{S}^{*}_{e} := \{f \in \mathcal{A} : zf'(z)/f(z) \prec e^z\}$, which denotes the class of starlike functions associated with the exponential function.

Keywords: Differential Subordination, Exponential Function, Starlike Function, Third-order Subordination

MSC numbers: 30C45, 30C80