New applications in third-order strong differential subordination theory

The research conducted in this investigation focuses on extending known results from the second-order differential subordination theory for the special case of third-order strong differential subordination. This paper intends to facilitate the development of new results in this theory by showing how specific lemmas used as tools in classical second-order differential subordination theory are adapted for the context of third-order strong differential subordination. Two theorems proved in this study extend two familiar lemmas due to D.J. Hallenbeck and S. Ruscheweyh, and G.M. Goluzin, respectively. A numerical example illustrates applications of the new results but the theorems are hoped to become helpful tools in generating new outcome for this very recently initiated line of research concerning third-order strong differential subordination.