scholarly journals Solvability of a New q-Differential Equation Related to q-Differential Inequality of a Special Type of Analytic Functions

2021 ◽  
Vol 5 (4) ◽  
pp. 228
Author(s):  
Ibtisam Aldawish ◽  
Rabha W. Ibrahim
Keyword(s):  
Differential Equation ◽  
Differential Operator ◽  
Analytic Functions ◽  
Differential Inequality ◽  
Laguerre Polynomial ◽  
Analytic Solutions ◽  
Confluent Hypergeometric Function ◽  
Differential Inequalities ◽  
Quantum Calculus

The current study acts on the notion of quantum calculus together with a symmetric differential operator joining a special class of meromorphic multivalent functions in the puncher unit disk. We formulate a quantum symmetric differential operator and employ it to investigate the geometric properties of a class of meromorphic multivalent functions. We illustrate a set of differential inequalities based on the theory of subordination and superordination. In this real case study, we found the analytic solutions of q-differential equations. We indicate that the solutions are given in terms of confluent hypergeometric function of the second type and Laguerre polynomial.

scholarly journals A New Subclass of Analytic Functions Defined by Using Salagean q-Differential Operator

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 458 ◽  
Author(s):  
Muhammad Naeem ◽  
Saqib Hussain ◽  
Tahir Mahmood ◽  
Shahid Khan ◽  
Maslina Darus
Keyword(s):  
Differential Operator ◽  
Analytic Functions ◽  
Extreme Points ◽  
Distortion Theorem ◽  
Coefficient Estimates ◽  
Quantum Calculus

In our present investigation, we use the technique of convolution and quantum calculus to study the Salagean q-differential operator. By using this operator and the concept of the Janowski function, we define certain new classes of analytic functions. Some properties of these classes are discussed, and numerous sharp results such as coefficient estimates, distortion theorem, radii of star-likeness, convexity, close-to-convexity, extreme points, and integral mean inequalities of functions belonging to these classes are obtained and studied.

scholarly journals Sufficient conditions for analytic functions defined by Frasin differential operator

2021 ◽  
Vol 66 (2) ◽  
pp. 353-359
Author(s):  
Tariq Al-Hawary
Keyword(s):  
Differential Operator ◽  
Complex Plane ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Current Work ◽  
Differential Inequalities

"Very recently, Frasin [7] introduced the differential operator $\mathcal{I}_{m,\lambda }^{\zeta }f(z)$ defined as \begin{equation*} \mathcal{I}_{m,\lambda }^{\zeta }f(z)=z+\sum\limits_{n=2}^{\infty }\left( 1+(n-1)\sum\limits_{j=1}^{m}\binom{m}{j}(-1)^{j+1}\lambda ^{j}\right) ^{\zeta }a_{n}z^{n}. \end{equation*} The current work contributes to give an application of the differential operator $\mathcal{I}_{m,\lambda }^{\zeta }f(z)$ to the differential inequalities in the complex plane."

Linear directional differential equations in the unit ball: solutions of bounded L-index

2019 ◽  
Vol 69 (5) ◽  
pp. 1089-1098 ◽  
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Unit Ball ◽  
Analytic Functions ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Analytic Solutions ◽  
Directional Derivatives ◽  
Several Variables

Abstract We study sufficient conditions of boundedness of L-index in a direction b ∈ ℂn ∖ {0} for analytic solutions in the unit ball of a linear higher order non-homogeneous differential equation with directional derivatives. These conditions are restrictions by the analytic coefficients in the unit ball of the equation. Also we investigate asymptotic behavior of analytic functions of bounded L-index in the direction and estimate its growth. The results are generalizations of known propositions for entire functions of several variables.

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 On a class of analytic functions associated to a complex domain concerning q-differential-difference operator

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Rabha W. Ibrahim ◽  
Maslina Darus
Keyword(s):  
Differential Operator ◽  
Analytic Functions ◽  
Existence Of Solutions ◽  
Difference Operator ◽  
Open Unit ◽  
Convolution Product ◽  
Open Unit Disk ◽  
Current Investigation ◽  
Quantum Calculus ◽  
New Class

AbstractIn our current investigation, we apply the idea of quantum calculus and the convolution product to amend a generalized Salagean q-differential operator. By considering the new operator and the typical version of the Janowski function, we designate definite new classes of analytic functions in the open unit disk. Significant properties of these modules are considered, and recurrent sharp consequences and geometric illustrations are realized. Applications are considered to find the existence of solutions of a new class of q-Briot–Bouquet differential equations.

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.

scholarly journals Geometric Inequalities via a Symmetric Differential Operator Defined by Quantum Calculus in the Open Unit Disk

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Rabha W. Ibrahim ◽  
Rafida M. Elobaid ◽  
Suzan J. Obaiys
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Unit Disk ◽  
Analytic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Geometric Inequalities ◽  
Quantum Calculus ◽  
Convolution Operation ◽  
Geometric Representations

The present investigation covenants with the concept of quantum calculus besides the convolution operation to impose a comprehensive symmetric q-differential operator defining new classes of analytic functions. We study the geometric representations with applications. The applications deliberated to indicate the certainty of resolutions of a category of symmetric differential equations type Briot-Bouquet.

scholarly journals Generalized Quantum Integro-Differential Fractional Operator with Application of 2D-Shallow Water Equation in a Complex Domain

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 342
Author(s):  
Rabha W. Ibrahim ◽  
Dumitru Baleanu
Keyword(s):  
Differential Operator ◽  
Shallow Water ◽  
Analytic Functions ◽  
Shallow Water Equation ◽  
Complex Domain ◽  
Open Unit ◽  
Open Unit Disk ◽  
Fractional Operator ◽  
Quantum Calculus ◽  
Generalized Operator

In this paper, we aim to generalize a fractional integro-differential operator in the open unit disk utilizing Jackson calculus (quantum calculus or q-calculus). Next, by consuming the generalized operator to define a formula of normalized analytic functions, we present a set of integral inequalities using the concepts of subordination and superordination. In addition, as an application, we determine the maximum and minimum solutions of the extended fractional 2D-shallow water equation in a complex domain.

Asymptotic Behaviour of Solutions of Parabolic Differential Inequalities

1962 ◽  
Vol 14 ◽  
pp. 626-631 ◽  
Author(s):  
Milton Lees
Keyword(s):  
Differential Operator ◽  
Boundary Conditions ◽  
Asymptotic Behaviour ◽  
Differential Inequality ◽  
Second Order ◽  
Differential Inequalities ◽  
Order Linear ◽  
Open Set ◽  
Asymptotic Behaviour Of Solutions ◽  
Image Position

Let there be given a parabolic differential operatorwhere A is a second order linear elliptic (<0) differential operator in an open set Ω ⊂ Rn, having coefficients depending on x ∈ Ω and t ∈ [0, ∞]. Recently, Protter (1) investigated the asymptotic behaviour of functions u(x, t) that satisfy the differential inequality(1.1)Under suitable restrictions on the functions ci(t) and the coefficients of A, he proved that any solution of (1.1), subject to certain homogeneous boundary conditions, that vanishes sufficiently fast, as t → ∞, must be identically zero in Ω × [0, ∞). For example, conditions are given under which no solution of (1.1) can vanish faster than e-λt, ∀ λ > 0, unless identically zero.

A NEW SUBCLASS OF ANALYTIC FUNCTIONS DEFINED BY OPOOLA DIFFERENTIAL OPERATOR

2020 ◽  
Vol 9 (8) ◽  
pp. 5343-5348 ◽  
Author(s):  
T. G. Shaba ◽  
A. A. Ibrahim ◽  
M. F. Oyedotun
Keyword(s):  
Differential Operator ◽  
Analytic Functions
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