Generalized Briot-Bouquet Differential Equation Based on New Differential Operator with Complex Connections

Rafida M. Elobaid,Suzan J. Obaiys,Rabha W. Ibrahim

doi:10.3390/axioms9020042

Abstract

A class of Briot–Bouquet differential equations is a magnificent part of investigating the geometric behaviors of analytic functions, using the subordination and superordination concepts. In this work, we aim to formulate a new differential operator with complex connections (coefficients) in the open unit disk and generalize a class of Briot–Bouquet differential equations (BBDEs). We study and generalize new classes of analytic functions based on the new differential operator. Consequently, we define a linear operator with applications.

Highlights

Inequalities in a complex domain play a massive role in function theory
They have been employed to introduce the geometric interpolation of analytic functions in the open unit disk
Lupas [1] suggested a combination of two famous differential operators given by Ruscheweyh [2] and Sàlàgean [3] to present a set of inequalities and inclusions by using the concept of subordination

Summary

Introduction

Inequalities in a complex domain play a massive role in function theory. They have been employed to introduce the geometric interpolation of analytic functions in the open unit disk. They have been utilized to formulate generalized classes of analytic functions. We shall define a new differential operator of complex coefficients and study its behaviors based on the properties of the theory of geometric functions. The new operator will be formulated in generalized sub-classes of starlike functions. We introduce a generalization of a class of Briot–Bouquet differential equations (BBDEs) in the complex domain. A comparison with recent works is shown in the sequel

Differential Operators

Briot–Bouquet Differential Equation

Linear Combination Operator

Subordination Inequalities

Conclusions