Derivative Operator of Order ε+ρ-1 Associated with Differential Subordination and Superordination

Abstract

Professors Miller and Mocanu established the theory of differential subordination and its twin, the theory of differential super ordination, which are both based on reinterpreting fundamental inequalities for real-valued functions for the situation of complex-valued functions. Using different types of operators to study subordination and super ordination characteristics is a technique that is still extensively employed, with some investigations leading to sandwich-type theorems, as is the case in the current work. The objective of this work is to derive differential Subordination and Super ordination outcomes using the derivative operator of order E+-1. Differential subordination and super ordination results are achieved for analytic functions connected with the integral operator in the open unit disc. These findings are achieved by examining relevant types of admissible functions, differential supremacy theorem, several operator differential hyperboloids requiring partial integration of a stacking suprageometric function are produced, as well as the best subordinates. The result of a sandwich type links the outcomes of dependency and dependency using Theorem 9. Keep track of intriguing corollaries for certain occupations by using the best subordinate and dominant skills. Presented in this paper may be used to motivate the usage of alternative hyper-geometric functions related to partial integration.