On Multivalent Analytic Functions Considered by a Multi-Arbitrary Differential Operator in a Complex Domain

Najla M Alarifi,Rabha W Ibrahim

doi:10.3390/axioms10040315

Najla M Alarifi, Rabha W Ibrahim

Open Access

https://doi.org/10.3390/axioms10040315

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Abstract

(1) Background: There is an increasing amount of information in complex domains, which necessitates the development of various kinds of operators, such as differential, integral, and linear convolution operators. Few investigations of the fractional differential and integral operators of a complex variable have been undertaken. (2) Methods: In this effort, we aim to present a generalization of a class of analytic functions based on a complex fractional differential operator. This class is defined by utilizing the subordination and superordination theory. (3) Results: We illustrate different fractional inequalities of starlike and convex formulas. Moreover, we discuss the main conditions to obtain sandwich inequalities involving the fractional operator. (4) Conclusion: We indicate that the suggested class is a generalization of recent works and can be applied to discuss the upper and lower bounds of a special case of fractional differential equations.

Highlights

Geometric function theory’s primary research objective is to introduce new classes of analytic functions and to explore their geometric shapes
(2) Methods: In this effort, we aim to present a generalization of a class of analytic functions based on a complex fractional differential operator
We aim to present two new classes of multivalent analytic function types based on the generalized fractional differential operator

Summary

Introduction

Geometric function theory’s primary research objective is to introduce new classes of analytic functions and to explore their geometric shapes. There are many classes of analytic functions in the open unit disk, such as normalized, multivalent, harmonic and meromorphic functions, formulating different geometric processes. These processes present a derivative, integral or convolution operationally—for example, the Salagean differential operator [1] and its generalizations [2,3], conformabale differential operator [4] and symmetric differential operator [5]. The most popular fractional operators are Riemann–Liouville fractional differential and integral operators These operators were extended to the complex plane by Owa and Srivastava [6] and generalized for 2D-parametric fractional power in [7,8]

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