Q-difference Operators Research Articles

Certain New Applications of Faber Polynomial Expansion for a New Class of bi-Univalent Functions Associated with Symmetric q-Calculus

In this study, we applied the ideas of subordination and the symmetric q-difference operator and then defined the novel class of bi-univalent functions of complex order γ. We used the Faber polynomial expansion method to determine the upper bounds for the functions belonging to the newly defined class of complex order γ. For the functions in the newly specified class, we further obtained coefficient bounds ρ2 and the Fekete–Szegő problem ρ3−ρ22, both of which have been restricted by gap series. We demonstrate many applications of the symmetric Sălăgean q-differential operator using the Faber polynomial expansion technique. The findings in this paper generalize those from previous studies.

On Sequence Spaces Due to lth Order q-Difference Operator and its Spectrum

On Sequence Spaces Due to lth Order q-Difference Operator and its Spectrum

Sharp Bounds of the Fekete–Szegö Problem and Second Hankel Determinant for Certain Bi-Univalent Functions Defined by a Novel q-Differential Operator Associated with q-Limaçon Domain

In this present paper, we define a new operator in conjugation with the basic (or q-) calculus. We then make use of this newly defined operator and define a new class of analytic and bi-univalent functions associated with the q-derivative operator. Furthermore, we find the initial Taylor–Maclaurin coefficients for these newly defined function classes of analytic and bi-univalent functions. We also show that these bounds are sharp. The sharp second Hankel determinant is also given for this newly defined function class.

Certain Class of Bi-Univalent Functions Defined by Sălăgean q-Difference Operator Related with Involution Numbers

We introduce and examine two new subclass of bi-univalent function Σ, defined in the open unit disk, based on Sălăgean-type q-difference operators which are subordinate to the involution numbers. We find initial estimates of the Taylor–Maclaurin coefficients |a2| and |a3| for functions in the new subclass introduced here. We also obtain a Fekete–Szegö inequality for the new function class. Several new consequences of our results are pointed out, which are new and not yet discussed in association with involution numbers.

The Properties of Meromorphic Multivalent q-Starlike Functions in the Janowski Domain

Many researchers have defined the q-analogous of differential and integral operators for analytic functions using the concept of quantum calculus in the geometric function theory. In this study, we conduct a comprehensive investigation to identify the uses of the Sălăgean q-differential operator for meromorphic multivalent functions. Many features of functions that belong to geometrically defined classes have been extensively studied using differential operators based on q-calculus operator theory. In this research, we extended the idea of the q-analogous of the Sălăgean differential operator for meromorphic multivalent functions using the fundamental ideas of q-calculus. With the help of this operator, we extend the family of Janowski functions by adding two new subclasses of meromorphic q-starlike and meromorphic multivalent q-starlike functions. We discover significant findings for these new classes, including the radius of starlikeness, partial sums, distortion theorems, and coefficient estimates.

Application of the Quasi-Hadamard Product to Subclasses of Analytic Functions Involving the q-Difference Operator

In this study, the subclasses Vq,ϱℓ(c𝚥,λ),Uq,ϱ(c𝚥,λ) and Vq,ϱ(c𝚥,λ) of analytic functions using the q-difference operator are defined and investigated. Thus, we obtained some results for the quasi-Hadamard product on these classes of analytic functions. Furthermore, connections between our results and some previously established results are outlined in this study.

On Generalizations of the Close-to-Convex Functions Associated with q-Srivastava–Attiya Operator

The study of the q-analogue of the classical results of geometric function theory is currently of great interest to scholars. In this article, we define generalized classes of close-to-convex functions and quasi-convex functions with the help of the q-difference operator. Moreover, by using the q-analogues of a certain family of linear operators, the classes Kq,bsh, K˜q,sbh, Qq,bsh, and Q˜q,sbh are introduced. Several interesting inclusion relationships between these newly defined classes are discussed, and the invariance of these classes under the q-Bernadi integral operator was examined. Furthermore, some special cases and useful consequences of these investigations were taken into consideration.

Linear 𝑞-difference, difference and differential operators preserving some 𝒜-entire functions

We apply Rossi’s half-plane version of Borel’s theorem to study the zero distribution of linear combinations of A \mathcal {A} -entire functions (Theorem 1.2). This provides a unified way to study linear q q -difference, difference and differential operators (with entire coefficients) preserving subsets of A \mathcal {A} -entire functions, and hence obtain several analogous results for the Hermite-Poulain theorem to linear finite ( q q -)difference operators with polynomial coefficients. The method also produces a result on the existence of infinitely many non-real zeros of some differential polynomials of functions in certain sub-classes of A \mathcal {A} -entire functions.

Some Subordination Results Defined by Using the Symmetric q-Differential Operator for Multivalent Functions

In this article, we use the concept of symmetric q-calculus and convolution in order to define a symmetric q-differential operator for multivalent functions. This operator is an extension of the classical Ruscheweyh differential operator. By using the technique of differential subordination, we derive several interesting applications of the newly defined operator for multivalent functions.

Sharp Coefficient Bounds for a New Subclass of q-Starlike Functions Associated with q-Analogue of the Hyperbolic Tangent Function

In this study, by making the use of q-analogous of the hyperbolic tangent function and a Sălăgean q-differential operator, a new class of q-starlike functions is introduced. The prime contribution of this study covers the derivation of sharp coefficient bounds in open unit disk U, especially the first three coefficient bounds, Fekete–Szego type functional, and upper bounds of second- and third-order Hankel determinant for the functions to this class. We also use Zalcman and generalized Zalcman conjectures to investigate the coefficient bounds of a newly defined class of functions. Furthermore, some known corollaries are highlighted based on the unique choices of the involved parameters l and q.

Some New Applications of the Faber Polynomial Expansion Method for Generalized Bi-Subordinate Functions of Complex Order γ Defined by q-Calculus

This work examines a new subclass of generalized bi-subordinate functions of complex order γ connected to the q-difference operator. We obtain the upper bounds ρm for generalized bi-subordinate functions of complex order γ using the Faber polynomial expansion technique. Additionally, we find coefficient bounds ρ2 and Feke–Sezgo problems ρ3−ρ22 for the functions in the newly defined class, subject to gap series conditions. Using the Faber polynomial expansion method, we show some results that illustrate diverse uses of the Ruschewey q differential operator. The findings in this paper generalize those from previous efforts by a number of prior researchers.

Pastro polynomials and Sobolev-type orthogonal polynomials on the unit circle based on a q-difference operator

ABSTRACT Applications of orthogonal polynomials on the unit circle have attracted the attention of many researchers in recent years. Pastro polynomials, which are basic hypergeometric polynomials, are known to be biorthogonal polynomials on the unit circle. However, with special choice of parameters they provide one of the nicest examples of orthogonal polynomials on the unit circle. Our objective here is to consider some properties of three sequences of polynomials which are related to these Pastro orthogonal polynomials on the unit circle by a q-difference operator. We also provide information regarding connection formulas, bounds for the connection coefficients as well as outer relative asymptotics associated with these sequences of polynomials.

New Applications of Faber Polynomial Expansion for Analytical Bi-Close-to-Convex Functions Defined by Using q-Calculus

In this investigation, the q-difference operator and the Sălăgean q-differential operator are utilized to establish novel subclasses of analytical bi-close-to-convex functions. We determine the general Taylor–Maclaurin coefficient of the functions in this class using the Faber polynomial method. We demonstrate the unpredictable behaviour of initial coefficients a2, a3 and investigate the Fekete–Szegő problem a3−a22 for the subclasses of bi-close-to-convex functions. To highlight the connections between existing knowledge and new research, certain known and unknown corollaries are also highlighted.

Certain Inclusion Properties for the Class of q-Analogue of Fuzzy α-Convex Functions

Recently, the properties of analytic functions have been mainly discussed by means of a fuzzy subset and a q-difference operator. We define certain new subclasses of analytic functions by using the fuzzy subordination to univalent functions whose range is symmetric with respect to the real axis. We introduce the family of linear q-operators and define various classes associated with these operators. The inclusion results and various integral properties are the main investigations of this article.

Subordination theorems for a class related to q-fractional differential operator

By using the definition of q-difference operator, we defined the new q-Al-Oboudi-Al-Amoudi operator, which generalize modified Al-Oboudi-Al-Amoudi operator. Using the new operator, we defined a new class of uniformly functions and obtained subordination result for functions in it. Our results not only generalize previous results but also modified some previous results.

Coefficient estimates for certain class of bi-univalent functions associated with κ-Fibonacci numbers

In the present work, we propose to introduce and investigate a new class SLΣq,μ (γ, λ, n, pκ) of the function class Σ of bi-univalent functions related to κ-Fibonacci numbers defined in the open unit disk, which is associated with the Salagean type q-difference operator and satisfy some subordination conditions. We obtain coefficient bounds for the Taylor–Maclaurin coefficients |a2| and |a3| of the functions in the new class. Furthermore, we solve the Fekete–Szegö functional for functions in the class SLΣq,μ (γ, λ, n, pκ).

Certain subclass of meromorphic univalent positive coefficients defined by q-difference operator

This paper introduces a Certain Subclass of Meromorphic Univalent Positives Coefficients Defined by the q-Difference Operator. Coefficient estimates are investigated and obtained, and the upped bound is calculated.

On fuzzy differential subordination associated with $ q $-difference operator

<abstract><p>This article presents the link between the fuzzy differential subordination and the q-theory of functions. We use the fuzzy differential subordination to define certain subclasses of univalent functions associated with the q-difference operator. Certain inclusion results are proved, and invariance of the $ q $-Bernardi integral operator for certain classes is discussed.</p></abstract>

On a Certain Subclass of p-Valent Analytic Functions Involving q-Difference Operator

This paper introduces and studies a new class of analytic p-valent functions in the open symmetric unit disc involving the Sălăgean-type q-difference operator. Furthermore, we present several interesting subordination results, coefficient inequalities, fractional q-calculus applications, and distortion theorems.

Charting the [formula omitted]-Askey scheme. II. The [formula omitted]-Zhedanov scheme

This is the second in a series of papers which intend to explore conceptual ways of distinguishing between families in the q-Askey scheme and uniform ways of parametrizing the families. For a system of polynomials pn(x) in the q-Askey scheme satisfying Lpn=hnpn with L a second order q-difference operator the q-Zhedanov algebra is the algebra generated by operators L and X (multiplication by x). It has two relations in which essentially five coefficients occur. Vanishing of one or more of the coefficients corresponds to a subfamily or limit family of the Askey–Wilson polynomials. An arrow from one family to another means that in the latter family one more coefficient vanishes. This yields the q-Zhedanov scheme given in this paper.The q-hypergeometric expression of pn(x) can be interpreted as an expansion of pn(x) in terms of certain Newton polynomials. In our previous paper (Contemporary Math. 780) we used Verde-Star’s clean parametrization of such expansions and we obtained a q-Verde-Star scheme, where vanishing of one or more of these parameters corresponds to a subfamily or limit family. The actions of the operators L and X on the Newton polynomials can be expressed in terms of the Verde-Star parameters, and thus the coefficients for the q-Zhedanov algebra can be expressed in terms of these parameters. There are interesting differences between the q-Verde-Star scheme and the q-Zhedanov scheme, which are discussed in the paper.