Symmetric Conformable Fractional Derivative of Complex Variables

Rabha W Ibrahim,Suzan J Obaiys,Rafida M Elobaid

doi:10.3390/math8030363

Rabha W Ibrahim, Suzan J Obaiys + Show 1 more

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https://doi.org/10.3390/math8030363

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Abstract

It is well known that the conformable and the symmetric differential operators have formulas in terms of the first derivative. In this document, we combine the two definitions to get the symmetric conformable derivative operator (SCDO). The purpose of this effort is to provide a study of SCDO connected with the geometric function theory. These differential operators indicate a generalization of well known differential operator including the Sàlàgean differential operator. Our contribution is to impose two classes of symmetric differential operators in the open unit disk and to describe the further development of these operators by introducing convex linear symmetric operators. In addition, by acting these SCDOs on the class of univalent functions, we display a set of sub-classes of analytic functions having geometric representation, such as starlikeness and convexity properties. Investigations in this direction lead to some applications in the univalent function theory of well known formulas, by defining and studying some sub-classes of analytic functions type Janowski function and convolution structures. Moreover, by using the SCDO, we introduce a generalized class of Briot–Bouquet differential equations to introduce, what is called the symmetric conformable Briot–Bouquet differential equations. We shall show that the upper bound of this class is symmetric in the open unit disk.

Highlights

IntroductionThe term Symmetry from Greek means arrangement and organization in measurements
The term Symmetry from Greek means arrangement and organization in measurements.In free language, it mentions a concept of harmonious and attractive proportion and equilibrium.In mathematics, it discusses an object that is invariant via certain transformation or rotation or scaling.In geometry, the object has symmetry if there is an operator or transformation that maps the object onto itself [1,2].Sàlàgean (1983) presented a differential operator for a class of analytic functions.Many sub-classes of analytic functions are studied using this operator
We present a new symmetric conformable derivative operator (SCDO) in the open unit disk

Summary

Introduction

The term Symmetry from Greek means arrangement and organization in measurements. In free language, it mentions a concept of harmonious and attractive proportion and equilibrium. Sàlàgean (1983) presented a differential operator for a class of analytic functions (see [3]). Many sub-classes of analytic functions are studied using this operator. Our investigation is to study classes of analytic functions by using the symmetric differential operator in a complex domain. Mathematics 2020, 8, 363 which is called a complex conformable differential operator. This operator is an extension of the Anderson–Ulness operator [11]. We present a new SCDO in the open unit disk We formulate it in some sub-classes of univalent functions. We generalize a class of Briot–Bouquet differential equations by using SCDO

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Applications