Third-order differential subordination and superordination results for p-valent analytic function involving fractional derivative operator

Abstract

We introduce a concept that generalizes the fractional calculus (differentiation and integration) in the complex domain using the Mellin transform. Special cases that lead to the well-known classical forms are discussed. A real-case example, along with a corresponding plot, is provided for illustration. The fractional derivative mentioned above is utilized to present applications to the differential subordination results of Antonino and Miller, as well as the differential superordination results of Tang et al. for -valent analytic functions, ultimately leading to sandwich-type results. Finally, we highlight the potential application of this topic in fluid mechanics.