Differential Subordination and Superordination Results Using Fractional Integral of Confluent Hypergeometric Function

Alina Alb Lupaş,Georgia Irina Oros

doi:10.3390/sym13020327

Abstract

Both the theory of differential subordination and its dual, the theory of differential superordination, introduced by Professors Miller and Mocanu are based on reinterpreting certain inequalities for real-valued functions for the case of complex-valued functions. Studying subordination and superordination properties using different types of operators is a technique that is still widely used, some studies resulting in sandwich-type theorems as is the case in the present paper. The fractional integral of confluent hypergeometric function is introduced in the paper and certain subordination and superordination results are stated in theorems and corollaries, the study being completed by the statement of a sandwich-type theorem connecting the results obtained by using the two theories.

Highlights

Superordination Results UsingThe theory of differential subordination emerged from the remark that using a realvalued function f twice continuously differentiable on an interval I = (−1, 1) and assuming that the differential operatorFractional Integral of ConfluentHypergeometric Function
Having as inspiration the results obtained by applying fractional integral on certain hypergeometric functions, the confluent hypergeometric function is considered in the present study and the fractional integral of confluent hypergeometric function is introduced
A new operator is defined and the theory of differential subordination is applied in order to obtain interesting subordinations related to it for which best dominants are given

Summary

Introduction

Many interesting outcomes of the study done using the theories of differential subordination and superordination are due to the use of operators. Conditions related to its univalence were stated in [10], its applications on certain classes of univalent functions are shown in [14] and an analytical study on Mittag-Leffler–confluent hypergeometric functions was conducted in [15] using fractional integral operator. Symmetry 2021, 13, 327 integral is introduced and studied using the theories of differential subordination and superordination of this paper. The univalent function q is called a dominant of the solutions of the differential subordination, or more a dominant, if p ≺ q for all p satisfying (6).

Results

Conclusion