Third order differential subordination associated with exponential function

UDC 517.5 Using admissibilty conditions, we investigate some third-order differential subordination results with a superordinate function $q(z)=e^z$. As an application, we obtain sufficient conditions for certain special functions to become exponential convex and exponential starlike. The derived second-order differential implications significantly generalize numerous results obtained in [A. Naz, S. Nagpal, V. Ravichandran, Mediterran. J. Math., 17, № 204, 1–22 (2020)].