Q-difference Operators Research Articles

Convolution Results with Subclasses of p-Valent Meromorphic Function Connected with q-Difference Operator

Applying the operator of q-difference, we examine the convolution properties of the subclasses MSζ,qr,p(A,B) and MKζ,qr,p(A,B) of p-valent meromorphic functions defined in the punctured open-unit disc. We derived specific inclusion features and coefficient estimates for functions that fall into these subclasses. Additionally, connections between the results presented here and those discovered in earlier papers are emphasized.

Some New Applications of the Quantum Calculus for New Families of Sigmoid Activation Bi-univalent Functions Connected to Horadam Polynomials

The study of q-calculus is becoming increasingly prominent in the field of geometric function theory, reflecting a growing interest in its applications. In this research work, we first develop a new type of modified Sigmoid-Salagean q-differential operator in the open unit disk D, utilizing the concepts of quantum calculus and the Sigmoid activation function. Using this newly defined q analogous differential operator and Horadam polynomials, we introduce new subclasses of bi-univalent functions in D. We determine upper bounds on initial coefficients, as well as the Fekete-Szeg ̈o problems, for functions belonging to these special families. Additionally, we discuss several interesting consequences related to the findings presented in this study.

Some Estimates for Certain q-analogs of Gamma Integral Transform Operators

The aim of this work is to examine some q-analogs and differential properties of the gamma integral operator and its convolution products. The q-gamma integral operator is introduced in two versions in order to derive pertinent conclusions regarding the q-exponential functions. Also, new findings on the q-trigonometric, q-sine, and q-cosine functions are extracted. In addition, novel results for first and second-order q-differential operators are established and extended to Heaviside unit step functions. Lastly, three crucial convolution products and extensive convolution theorems for the q-analogs are also provided.

Some Coefficient Problems for Subclasses of Holomorphic Functions in Complex Order Associating Sălăgeăn q-Differential Operator

A function with complex values and at every point of the specific domain contains a derivative is commonly known as analytic functions which is also referred as holomorphic functions. We begin by interpreting \(A\) as the class for all holomorphic functions \(L(\xi)\) with a Taylor series expansion written in the form: \[L(\xi) = \xi + \sum_{i=2}^{\infty} x_i \xi^i\] where \(x_i \in \mathbb{C}\) and \(\xi \in D\). \(D\) is the open unit disk where \(D = \{\xi : \xi \in \mathbb{C}, |\xi| < 1 \}\). Furthermore, we suggest the subclass of \(A\) that is univalent in \(D\) represented as \(S\). It is commonly known that the main subclasses of class \(S\) are the class of starlike functions and the class of convex functions. To develop and analyze the Fekete-Szegö problems, the theory of geometric function contributes significantly to this. Moreover, the frequent use of \(q\)-calculus as a general direction of research among mathematicians has caught our attention. In this research, we attained the initial coefficients, \(x_2\) and \(x_3\), and the upper bound for the functional \(|x_3 - \nu x_2^2|\) of functions \(L\) in the two new subclasses that are introduced by involving the Sălăgeăn \(q\)-differential operator, \(M_q^\eta L(\xi)\) and the definition of subordination.

New subclass of meromorphic harmonic functions defined by symmetric q-calculus and domain of Janowski functions

In this article, by applying the convolution principle and symmetric q-calculus, we develop a new generalized symmetric q-difference operator of convolution type, which is applicable in the domain E⁎={τ:τ∈C and 0<|τ|<∞}. Utilizing this operator, we construct, analyze, and evaluate two new sets of meromorphically harmonic functions in the Janowski domain. Furthermore, we investigate the convolution properties and necessary conditions for a function F to belong to the class MSHP,R(q,q−1), examining the sufficiency conditions for F to satisfy these properties. Moreover, we examine key geometric properties of the function F in the class MSH‾P,R(q,q−1), including the distortion bound, convex combinations, the extreme point theorem, and weighted mean estimates.

Sharp Results for a New Class of Analytic Functions Associated with the q-Differential Operator and the Symmetric Balloon-Shaped Domain

In our current study, we apply differential subordination and quantum calculus to introduce and investigate a new class of analytic functions associated with the q-differential operator and the symmetric balloon-shaped domain. We obtain sharp results concerning the Maclaurin coefficients the second and third-order Hankel determinants, the Zalcman conjecture, and its generalized conjecture for this newly defined class of q-starlike functions with respect to symmetric points.

Convolution Properties of Meromorphic P-Valent Functions with Coefficients of Alternating Type Defined Using q-Difference Operator

Convolution Properties of Meromorphic P-Valent Functions with Coefficients of Alternating Type Defined Using q-Difference Operator

Boundary values of Hankel and Toeplitz determinants for [formula omitted]-convex functions

The study of holomorphic functions has been recently extended through the application of diverse techniques, among which quantum calculus stands out due to its wide-ranging applications across various scientific disciplines. In this context, we introduce a novel q-differential operator defined via the generalized binomial series, which leads to the derivation of new classes of quantum-convex (q-convex) functions. Several specific instances within these classes were explored in detail. Consequently, the boundary values of the Hankel determinants associated with these functions were analyzed. All graphical representations and computational analyses were performed using Mathematica 12.0.•These classes are defined by utilizing a new q-differential operator.•The coefficient values |ai|(i=2,3,4) are investigated.•Toeplitz determinants, such as the second T2(2) and the third T3(1) order inequalities, are calculated.

On Common Borel Direction of Entire Function f and Its q-Difference Operator

The common Borel direction of entire function f(z) and its q-difference operator is studied by using Nevanlinna theory. Some conditions of the existence of common Borel direction for entire function f(z) and its q-difference operator are obtained in this paper.

Fractional Differential Operator Based on Quantum Calculus and Bi-Close-to-Convex Functions

In this article, we first consider the fractional q-differential operator and the λ,q-fractional differintegral operator Dqλ:A→A. Using the λ,q-fractional differintegral operator, we define two new subclasses of analytic functions: the subclass S*q,β,λ of starlike functions of order β and the class CΣλ,qα of bi-close-to-convex functions of order β. We explore the results on coefficient inequality and Fekete–Szegö problems for functions belonging to the class S*q,β,λ. Using the Faber polynomial technique, we derive upper bounds for the nth coefficient of functions in the class of bi-close-to-convex functions of order β. We also investigate the erratic behavior of the initial coefficients in the class CΣλ,qα of bi-close-to-convex functions. Furthermore, we address some known problems to demonstrate the connection between our new work and existing research.

Majorizations for subclasses of analytic functions connected with the Q-difference operator

Majorizations for subclasses of analytic functions connected with the Q-difference operator

Fekete-Szegӧ Functional for Classes X_q^n (φ) and Y_q^n (φ)

Two new subclasses of analytic functions are proposed by applying q-differential operator which is denoted as . Throughout this study, we acquired the initial coefficients and and the upper bound for the functional of the functions in the classes and .

Certain Subclasses of Bi-univalent Functions Defined by Linear Multiplier Fractional q-Differential Operator

This paper introduces a novel subclass of analytic and bi-univalent functions that are linked to a linear multiplier fractional q-differential operator, defined in the open unit disk D. The authors establish the upper bounds for the coefficients |a2| and |a3| for the functions that belong to this new subclass and its subclasses.

Some properties of a class of generalized Janowski-type $q$-starlike functions associated with Opoola $q$-differential operator and $q$-differential subordination

Without qualms, studies show that quantum calculus has received great attention in recent times. This can be attributed to its wide range of applications in many science areas. In this exploration, we study a new qdifferential operator that generalized many known differential operators. The new q-operator and the concept of subordination were afterwards, used to define a new subclass of analytic-univalent functions that invariably consists of several known and new generalizations of starlike functions. Consequently, some geometric properties of the new class were investigated. The properties include coefficient inequality, growth, distortion and covering properties. In fact, we solved some radii problems for the class and also established its subordinating factor sequence property. Indeed, varying some of the involving parameters in our results led to some existing results.

Fekete-Szegö Inequality for a Subclass of Bi-univalent Functions by Applying Sălăgean q-Differential Operator

Throughout this study, we propose a new subclass of bi-univalent functions by applying Sălăgean q-differential operator and denoted as LΣ_q^k (λ,ϕ). Further, we acquired the values of the initial coefficients |a_2 | and |a_3 | for functions f∈LΣ_q^k (λ,ϕ) which yield to this study’s preliminary result. Subsequently, the preliminary result was applied to obtain the upper bound of Fekete-Szegö inequality, |a_3-ρa_2^2 |, for functions f∈LΣ_q^k (λ,ϕ).

Sharp Estimates Involving a Generalized Symmetric Sălăgean q-Differential Operator for Harmonic Functions via Quantum Calculus

In this study, we apply q-symmetric calculus operator theory and investigate a generalized symmetric Sălăgean q-differential operator for harmonic functions in an open unit disk. We consider a newly defined operator and establish new subclasses of harmonic functions in complex order. We determine the sharp results, such as the sufficient necessary coefficient bounds, the extreme of closed convex hulls, and the distortion theorems for a new family of harmonic functions. Further, we illustrate how we connect the findings of previous studies and the results of this article.

Properties for a Certain Subclass of Analytic Functions Associated with the Salagean q-Differential Operator

Using the Salagean q-differential operator, we investigate a novel subclass of analytic functions in the open unit disc, and we use the Hadamard product to provide some inclusion relations. Furthermore, the coefficient conditions, convolution properties, and applications of the q-fractional calculus operators are investigated for this class of functions. In addition, we extend the Miller and Mocanu inequality to the q-theory of analytic functions.

On Coefficient Inequalities of Starlike Functions Related to the q-Analog of Cosine Functions Defined by the Fractional q-Differential Operator

This article extends the study of q-versions of analytic functions by introducing and studying the association of starlike functions with trigonometric cosine functions, both defined in their q-versions. Certain coefficient inequalities like coefficient bounds, Zalcman inequalities, and both Hankel and Toeplitz determinants for the new version of starlike functions are investigated. It is worth mentioning that most of the determined inequalities are sharp with the support of relevant extremal functions.

On a Subclass of Meromorphic multivalent Functions associated with q-differential operator

In this present paper, we have introduced and defined a new subclass of meromorphic multivalent function using the q-differential operator in punctured unit disk We have obtained some geometric properties like coefficient inequality, a sufficient and necessary condition for the meromorphic multivalent functions to be in the class and we have get a corollary from that theorem. Also, we have acquired the growth and distortion theorems bounds for the class Subsequently we have fixed the radii of starlikeness and convexity theorems. After that we have proved the class is closed under convex linear combinations. Also, we have introduced the extreme points theorem and proved that. Finally, we have obtained arithmetic mean theorem in the same class.

Applications of the Symmetric Quantum-Difference Operator for New Subclasses of Meromorphic Functions

Our goal in this article is to use ideas from symmetric q-calculus operator theory in the study of meromorphic functions on the punctured unit disc and to propose a novel symmetric q-difference operator for these functions. A few additional classes of meromorphic functions are then defined in light of this new symmetric q-difference operator. We prove many useful conclusions regarding these newly constructed classes of meromorphic functions, such as convolution, subordination features, integral representations, and necessary conditions. The technique presented in this article may be used to produce a wide variety of new types of generalized symmetric q-difference operators, which can subsequently be used to investigate a wide variety of new classes of analytic and meromorphic functions related to symmetric quantum calculus.