The objective of this paper is to examine the outcomes of third-order differential subordination and superordination for analytic functions within the region , specifically focusing on the utilization of the novel Hadamard operator . The results are derived by analyzing relevant categories of allowable functions. New findings have been found about differential subordination and superordination, along with the discovery of several sandwich theorems. Furthermore, other specific instances are also seen. The qualities and outcomes of differential subordination exhibit symmetry with the properties of differential superordination, leading to the formulation of the sandwich theorems.
Shakir et al. (Wed,) studied this question.