New Symmetric Differential and Integral Operators Defined in the Complex Domain

Maslina Darus,Rabha W Ibrahim

doi:10.3390/sym11070906

Abstract

The symmetric differential operator is a generalization operating of the well-known ordinary derivative. These operators have advantages in boundary value problems, statistical studies and spectral theory. In this effort, we introduce a new symmetric differential operator (SDO) and its integral in the open unit disk. This operator is a generalization of the Sàlàgean differential operator. Our study is based on geometric function theory and its applications in the open unit disk. We formulate new classes of analytic functions using SDO depending on the symmetry properties. Moreover, we define a linear combination operator containing SDO and the Ruscheweyh derivative. We illustrate some inclusion properties and other inequalities involving SDO and its integral.

Highlights

IntroductionInvestigation of the theory of operators (differential, integral, mixed, convolution and linear) has been a capacity of apprehension for numerous scientists in all fields of mathematical sciences, such as mathematical physics, mathematical biology and mathematical computing
Investigation of the theory of operators has been a capacity of apprehension for numerous scientists in all fields of mathematical sciences, such as mathematical physics, mathematical biology and mathematical computing
An additional definite field is the study of inequalities in the complex domain

Summary

Introduction

Investigation of the theory of operators (differential, integral, mixed, convolution and linear) has been a capacity of apprehension for numerous scientists in all fields of mathematical sciences, such as mathematical physics, mathematical biology and mathematical computing. The relationship of geometry and analysis signifies a very central feature in geometric function theory in the open unit disk. In 1983, Sàlàgean introduced his famous differential operator of normalized analytic functions in the open unit disk [2]. This operator is generalized and extended to many classes of univalent functions. It plays a significant tool to develop the geometric structure of many analytic functions by suggesting different classes Later this operator has been generalized and motivated by many researchers, for example, the Al-Oboudi differential operator [3]. Our research is to formulate a new symmetric differential operator and its integral by utilizing the concept of the symmetric derivative of complex variables This concept is an operation, extending the original derivative. Such methods enable methods for branching solutions construction using group symmetry [5,6]

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