scholarly journals Applications of Fractional Derivative on a Differential Subordinations and Superordinators for Analytic Functions Associated with Differential Operator

2021 ◽  
Vol 45 (03) ◽  
pp. 379-392
Author(s):  
ABBAS KAREEM WANAS ◽  
MASLINA DARUS
Keyword(s):  
Differential Operator ◽  
Fractional Derivative ◽  
Unit Disk ◽  
Analytic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Differential Subordinations

The purpose of this paper is to derive subordination and superordination results involving fractional derivative of differential operator for analytic functions in the open unit disk. These results are applied to obtain sandwich results. Our results extend corresponding previously known results.

scholarly journals Differential subordinations and superordinations for p-valent functions defined by fractional derivative operator

2013 ◽  
Vol 46 (3) ◽  
Author(s):  
Somia Muftah Amsheri ◽  
Valentina Zharkova
Keyword(s):  
Fractional Derivative ◽  
Unit Disk ◽  
Open Unit ◽  
Open Unit Disk ◽  
Derivative Operator ◽  
Differential Subordinations

AbstractIn the present paper, we derive some subordination and superordination results for p-valent functions in the open unit disk by using certain fractional derivative operator. Relevant connections of the results, which are presented in the paper, with various known results are also considered.

LIMITS OF FRACTIONAL DERIVATIVES AND COMPOSITIONS OF ANALYTIC FUNCTIONS

2016 ◽  
Vol 103 (1) ◽  
pp. 104-115 ◽  
Author(s):  
THOMAS H. MACGREGOR ◽  
MICHAEL P. STERNER
Keyword(s):  
Fractional Derivative ◽  
Unit Disk ◽  
Power Function ◽  
Complex Plane ◽  
Analytic Functions ◽  
Fractional Derivatives ◽  
Hadamard Product ◽  
Open Unit ◽  
Open Unit Disk

Suppose that the function $f$ is analytic in the open unit disk $\unicode[STIX]{x1D6E5}$ in the complex plane. For each $\unicode[STIX]{x1D6FC}>0$ a function $f^{[\unicode[STIX]{x1D6FC}]}$ is defined as the Hadamard product of $f$ with a certain power function. The function $f^{[\unicode[STIX]{x1D6FC}]}$ compares with the fractional derivative of $f$ of order $\unicode[STIX]{x1D6FC}$. Suppose that $f^{[\unicode[STIX]{x1D6FC}]}$ has a limit at some point $z_{0}$ on the boundary of $\unicode[STIX]{x1D6E5}$. Then $w_{0}=\lim _{z\rightarrow z_{0}}f(z)$ exists. Suppose that $\unicode[STIX]{x1D6F7}$ is analytic in $f(\unicode[STIX]{x1D6E5})$ and at $w_{0}$. We show that if $g=\unicode[STIX]{x1D6F7}(f)$ then $g^{[\unicode[STIX]{x1D6FC}]}$ has a limit at $z_{0}$.

scholarly journals Fractional heat equation optimized by a chaotic function

2021 ◽  
Vol 25 (Spec. issue 2) ◽  
pp. 173-178
Author(s):  
Rabha Ibrahim ◽  
Mayada Wazi ◽  
Dumitru Baleanu ◽  
Nadia Al-Saidi
Keyword(s):  
Differential Operator ◽  
Heat Equation ◽  
Unit Disk ◽  
Analytic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Fractional Heat Equation ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Extended Operator

In this effort, we propose a new fractional differential operator in the open unit disk. The operator is an extension of the Atangana-Baleanu differential operator without singular kernel. We suggest it for a normalized class of analytic functions in the open unit disk. By employing the extended operator, we study the time-2-D space heat equation and optimizing its solution by a chaotic function.

scholarly journals A New Parametric Differential Operator of p-Valently Analytic Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Samir B. Hadid ◽  
Rabha W. Ibrahim ◽  
G. Murugusundaramoorthy
Keyword(s):  
Differential Operator ◽  
Unit Disk ◽  
Analytic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Geometric Properties

Newly, numerous investigations are considered utilizing the idea of parametric operators (integral and differential). The objective of this effort is to formulate a new 2D-parameter differential operator (PDO) of a class of multivalent functions in the open unit disk. Consequently, we formulate the suggested operator in some interesting classes of analytic functions to study its geometric properties. The recognized class contains some recent works.

scholarly journals Sandwich Theorems for Multivalent Analytic Functions Associated with Differential Operator

2021 ◽  
Vol 45 (01) ◽  
pp. 7-20
Author(s):  
ABBAS KAREEM WANAS ◽  
ALB LUPAŞ ALINA
Keyword(s):  
Differential Operator ◽  
Unit Disk ◽  
Analytic Functions ◽  
Open Unit ◽  
Open Unit Disk

The purpose of this paper is to derive subordination and superordination results involving differential operator for multivalent analytic functions in the open unit disk. These results are applied to obtain sandwich results. Our results extend corresponding previously known results.

scholarly journals Symmetric Conformable Fractional Derivative of Complex Variables

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 363 ◽  
Author(s):  
Rabha W. Ibrahim ◽  
Rafida M. Elobaid ◽  
Suzan J. Obaiys
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Unit Disk ◽  
Analytic Functions ◽  
Differential Operators ◽  
Function Theory ◽  
Univalent Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Symmetric Differential Operators

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.

scholarly journals Study on subclasses of analytic functions

2017 ◽  
Vol 9 (1) ◽  
pp. 122-139 ◽  
Author(s):  
Imran Faisal ◽  
Maslina Darus
Keyword(s):  
Differential Operator ◽  
Unit Disk ◽  
Analytic Functions ◽  
Conjugate Points ◽  
Open Unit ◽  
Open Unit Disk ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Convolution Properties

AbstractBy making use of new linear fractional differential operator, we introduce and study certain subclasses of analytic functions associated with Symmetric Conjugate Points and defined in the open unit disk 𝕌 = {z : |z| < 1}. Inclusion relationships are established and convolution properties of functions in these subclasses are discussed.

scholarly journals Generalized Briot-Bouquet differential equation based on new differential operator with complex connections

2020 ◽  
Vol 28 (1) ◽  
pp. 105-114
Author(s):  
Rabha W. Ibrahim
Keyword(s):  
Differential Equation ◽  
Differential Operator ◽  
Unit Disk ◽  
Analytic Functions ◽  
Difference Operator ◽  
Open Unit ◽  
Open Unit Disk ◽  
New Class ◽  
Generalized Operator ◽  
Generalized Differential

AbstractInequality study is a magnificent field for investigating the geometric behaviors of analytic functions in the open unit disk calling the subordination and superordination. In this work, we aim to formulate a generalized differential-difference operator. We introduce a new class of analytic functions having the generalized operator. Some subordination results are included in the sequel.

scholarly journals Double Integral Operators Concerning Starlike of Orderβ

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
Kazuo Kuroki ◽  
Shigeyoshi Owa
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Integral Transform ◽  
Integral Operators ◽  
Open Unit ◽  
Open Unit Disk ◽  
Differential Inequalities ◽  
Double Integral ◽  
Differential Subordinations

Double integral operators which were considered by S. S. Miller and P. T. Mocanu (Integral Transform. Spec. Funct.19(2008), 591–597) are discussed. In order to show the analytic functionf(z)is starlike of orderβin the open unit disk&#x1D54C;, the theory of differential subordinations for analytic functions is applied. The object of the present paper is to discuss some interesting conditions forf(z)to be starlike of orderβin&#x1D54C;concerned with second-order differential inequalities and double integral operators.

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

Axioms ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 42 ◽  
Author(s):  
Rabha W. Ibrahim ◽  
Rafida M. Elobaid ◽  
Suzan J. Obaiys
Keyword(s):  
Differential Equation ◽  
Differential Operator ◽  
Linear Operator ◽  
Differential Equations ◽  
Unit Disk ◽  
Analytic Functions ◽  
Open Unit ◽  
Open Unit Disk

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.

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