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The present study aims to investigate a majorization problem for a class of q-starlike functions and for the subclass S ∗ (χ) of starlike functions as well as to rectify certain previous results. For this goal, the valid form connected with Nehari's inequality for q-difference operators was presented. Further, our findings contribute to the novelty of the results of this topic by revealing new insights. In addition, our results with previous works have connected, drawing inspiration from existing papers. Also, several new implications of the main result expressed as corollaries were provided and advances in the studied topic were made. [ABSTRACT FROM AUTHOR]

Abstract Recently, El-Deeb and Cotîrlă (Mathematics 11:11234834, 2023) used the error function together with a q-convolution to introduce a new operator. By means of this operator the following class R α , ϒ λ , q ( δ , η ) $\mathcal{R}_{\alpha ,\Upsilon}^{\lambda ,q}(\delta ,\eta )$ of analytic functions was studied: R α , ϒ λ , q ( δ , η ) : = { F : ℜ ( ( 1 − δ + 2 η ) H ϒ λ , q F ( ζ ) ζ + ( δ − 2 η ) ( H ϒ λ , q F ( ζ ) ) ′ + η ζ ( H ϒ λ , q F ( ζ ) ) ″ ) } > α ( 0 ≦ α < 1 ) . $$\begin{aligned} &\mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta ,\eta ) \\ &\quad := \biggl\{ \mathcal{ F}: {\Re} \biggl( (1-\delta +2\eta ) \frac{\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta )}{\zeta}+(\delta -2\eta ) \bigl(\mathcal{H} _{\Upsilon}^{\lambda ,q}\mathcal{F}(\zeta ) \bigr) ^{{ \prime}}+\eta \zeta \bigl( \mathcal{H}_{\Upsilon}^{\lambda ,q} \mathcal{F}( \zeta ) \bigr) ^{{{\prime \prime}}} \biggr) \biggr\} \\ &\quad >\alpha \quad (0\leqq \alpha < 1). \end{aligned}$$ For these general analytic functions F ∈ R β , ϒ λ , q ( δ , η ) $\mathcal{F}\in \mathcal{R}_{\beta ,\Upsilon}^{\lambda ,q}(\delta , \eta )$ , we give upper bounds for the Fekete–Szegö functional and for the second and third Hankel determinants.

Recently, El-Deeb and Cotîrlă (Mathematics 11:11234834, 2023) used the error function together with a q-convolution to introduce a new operator. By means of this operator the following class R α , ϒ λ , q (δ , η) of analytic functions was studied: R α , ϒ λ , q (δ , η) : = { F : ℜ ((1 − δ + 2 η) H ϒ λ , q F (ζ) ζ + (δ − 2 η) (H ϒ λ , q F (ζ)) ′ + η ζ (H ϒ λ , q F (ζ)) ″) } > α (0 ≦ α < 1). For these general analytic functions F ∈ R β , ϒ λ , q (δ , η) , we give upper bounds for the Fekete–Szegö functional and for the second and third Hankel determinants. [ABSTRACT FROM AUTHOR]

An analytic function, also known as a holomorphic function, is a complex-valued function that is differentiable at every point within a given domain. In other words, a function f (z) is analytic in a domain U if it has a derivative f′(z) at every point z in U. Let A represent the set of functions f that are analytic within the open unit disk D = {z ∈ ℂ : |z| < 1}. These functions possess a normalized Taylor-Maclaurin series expansion written in the form f (z) = z + Í∞ n=2 an z n where an ∈ ℂ, n = 2, 3, . . .. In recent years, the field of q-calculus has gained significant attention and research interest among mathematicians. The applications of this field are broadly applied in numerous subdivisions of physics and mathematics. In this research, we assume that S∗q and ℝq are subclasses of analytic functions obtained by applying the q-derivative operator. The objective of this paper is to obtain estimates for coefficient inequalities and Toeplitz determinants whose elements are the coefficients an for f ∈ S∗q and f ∈ Rq .

Orthogonal q-polynomials, both new and old, have witnessed a huge and revived attention in recent years, because of their applications in many diverse areas of mathematics and other sciences. In Geometric Function Theory, different subclasses of analytic and bi-univalent functions have been investigated and studied involving different orthogonal q-polynomials. In our present investigation, motivated by these recent research going on, first, we define some new subclasses of q-bi-starlike functions with the help of certain q-derivative operator which involving the generalized q-Lommel polynomials and q-Chebyshev polynomials. We then obtain the initial coefficients bounds for our defined function classes. Furthermore, the Fekete–Szegö inequalities are obtained for these defined function classes.

This research presents a new group of mathematical functions connected to Bernoulli's Lemniscate, using the q-derivative. Expanding on previous studies, the research concentrates on determining coefficient approximations, the Fekete-Szego functional, Zalcman inequality, Krushkal inequality, along with the second and third Hankel determinants for this recently established collection of functions. Additionally, the study derives the Fekete-Szego inequality for the function ξf(ξ) and obtains the inverse function f−1(ξ) for this specific class. This research advances our understanding in this area and suggests for further exploration.

In this paper we obtain coefficient characterization, extreme points and distortion bounds for the classes of p - valent harmonic functions defined by q - derivative operator. Some of our results improve and generalize previously known results. [ABSTRACT FROM AUTHOR]

Considering the most recent developments in quantum calculus and its numerous uses in physics and applied mathematics, this topic has inspired and fascinated many investigators in a variety of mathematical areas. The primary goal of the current paper is to investigate majorization issue for a category of q-starlike functions. Moreover, some new consequences of main result, which was stated in the form of corollaries, were presented. [ABSTRACT FROM AUTHOR]

In this article, we represent A as the of analytic functions in the open unit disk. Further, new subclasses of analytic functions of complex order utilising q-derivative operator are generated. The subclasses are symbolised by Hq,b(ϕ) and Iq,b(ϕ). Additionally, we discover that these function classes are implicated with the Fekete-Szegö inequalities.

In this article, we use the q-derivative operator and the principle of subordination to define a new subclass of analytic functions related to the q-Ruscheweyh operator. Sufficient conditions, sharp bounds for the initial coefficients, a Fekete–Szegö functional and a Toeplitz determinant are investigated for this new class of functions. Additionally, we also present several established consequences derived from our primary findings. [ABSTRACT FROM AUTHOR]

The potential for widespread applications of the geometric and mapping properties of functions of a complex variable has motivated this article. On the other hand, the basic or quantum (or q-) derivatives and the basic or quantum (or q-) integrals are extensively applied in many different areas of the mathematical, physical and engineering sciences. Here, in this article, we first apply the q-calculus in order to introduce the q-derivative operator S η , p , q n , m . Secondly, by means of this q-derivative operator, we define an interesting subclass T ℵ λ , p n , m (η , α , κ) of the class of normalized analytic and multivalent (or p-valent) functions in the open unit disk U. This p-valent analytic function class is associated with the class κ - UCV of κ -uniformly convex functions and the class κ - UST of κ -uniformly starlike functions in U. For functions belonging to the normalized analytic and multivalent (or p-valent) function class T ℵ λ , p n , m (η , α , κ) , we then investigate such properties as those involving (for example) the coefficient bounds, distortion results, convex linear combinations, and the radii of starlikeness, convexity and close-to-convexity. We also consider a number of corollaries and consequences of the main findings, which we derived herein. [ABSTRACT FROM AUTHOR]

In this article, we define the $ q $-difference operator and Salagean $ q $-differential operator for $ \upsilon $-fold symmetric functions in open unit disk $ \mathcal{U} $ by first applying the concepts of $ q $-calculus operator theory. Then, we considered these operators in order to construct new subclasses for $ \upsilon $-fold symmetric bi-univalent functions. We establish the general coefficient bounds $ |a_{\upsilon k+1}| $ for the functions in each of these newly specified subclasses using the Faber polynomial expansion method. Investigations are also performed on Feketo-Sezego problems and initial coefficient bounds for the function $ h $ that belong to the newly discovered subclasses. To illustrate the relationship between the new and existing research, certain well-known corollaries of our main findings are also highlighted.

In this article, we define the q-difference operator and Salagean q-differential operator for v-fold symmetric functions in open unit disk U by first applying the concepts of q-calculus operator theory. Then, we considered these operators in order to construct new subclasses for v-fold symmetric bi-univalent functions. We establish the general coefficient bounds |avk+1| for the functions in each of these newly specified subclasses using the Faber polynomial expansion method. Investigations are also performed on Feketo-Sezego problems and initial coefficient bounds for the function h that belong to the newly discovered subclasses. To illustrate the relationship between the new and existing research, certain well-known corollaries of our main findings are also highlighted. [ABSTRACT FROM AUTHOR]

A new subclass TXq[λ,A,B] of analytic functions is introduced by making use of the q-derivative operator associated with the Pascal distribution. Certain properties of analytic functions in the subclass TXq[λ,A,B] are derived. Some known results are generalized.

In mathematical analysis, the q-analogue of a function refers to a modified version of the function that is derived from q-series expansions. This paper is focused on the q-analogue of the exponential function and investigates a class of convex functions associated with it. The main objective is to derive precise inequalities that bound the coefficients of these convex functions. In this research, the initial coefficient bounds, Fekete–Szegő problem, second and third Hankel determinant have been determined. These coefficient bounds provide valuable information about the behavior and properties of the functions within the considered class.

LetAdenote the class of functionsfwhich are analytic in the open unit diskU. The subclass ofAconsisting of univalent functionsis denoted byM. In this paper, we also consider a subclass ofMwhich is denoted byV, consisting of functions with negativecoefficients. In addition, this paper also studies the q-derivative operator. By combining the ideas, this paper introduced threesubclasses ofAwith negative coefficients involving q-derivative. Furthermore, the coefficient estimates, growth results and extremepoints were obtained for all of these classes.

In the present paper we introduce two new sub-categories MLq,η*(p, s; d) and MKq,η(p, s; d) for a variety of meromorphic operations using a q-derivative operator defined on a perforated unit disk. We use Cassinian Oval √1+ dz with d ∈ (0, 1] as a subordinant function. We also find the necessary and sufficient conditions for the activities of these clauses. [ABSTRACT FROM AUTHOR]

We introduce a new class of bi-univalent functions defined in the open unit disc and connected with a q-convolution. We find estimates for the general Taylor-Maclaurin coefficients of the functions in this class by using Faber polynomial expansions and we obtain an estimation for the Fekete-Szegö problem for this class. [ABSTRACT FROM AUTHOR]

In this paper we determine certain coefficient inequalities for the classes of q-starlike and q-convex function and find the sufficient conditions for generalized Bessel function to belonging in these classes. [ABSTRACT FROM AUTHOR]

By the principle of differential subordination and the $ q $-derivative operator, we introduce the $ q $-analog $ \mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma) $ of certain class of analytic functions associated with the generalized Pascal snail. Firstly, we obtain the coefficient estimates and Fekete-Szegö functional inequalities for this class. Meanwhile, we also estimate the corresponding symmetric Toeplitz determinant. Secondly, for all the above results we provide the corresponding results for the reduced classes $ \mathcal{SP}^{q}_{snail}(\alpha, \beta, \gamma) $ and $ \mathcal{RP}^{q}_{snail}(\alpha, \beta, \gamma) $. Thirdly, we characterize the Bohr radius problems for the function class $ \mathcal{SP}^{q}_{snail}(\alpha, \beta, \gamma) $. Lastly, we establish certain results for some new subclasses of functions defined by the neutrosophic Poisson distribution series.

In the present investigation we introduce two new subclasses MS*q( p,b; c) and MKq(p,b; c) of meromorphic multivalent functions by using q-derivative operator defined in the punctured unit disc. We use the Cassinian Oval √(1+cz) with c∈(0,1] as a subordinant function in this treatise. Also we derive several properties including convolution properties, the necessary and sufficient condition and coefficient estimates for these subclasses.

In this article, we use the q-derivative operator and the principle of subordination to define a new subclass of analytic functions related to the q-Ruscheweyh operator. Sufficient conditions, sharp bounds for the initial coefficients, a Fekete–Szegö functional and a Toeplitz determinant are investigated for this new class of functions. Additionally, we also present several established consequences derived from our primary findings.

In this paper, we introduce a new class of bi-univalent functions defined in the open unit disc and connected with a q-convolution. We find estimates for the general Taylor-Maclaurin coefficients of the functions in this class by using Faber polynomial expansions, and we obtain an estimation for Fekete-Szegö problem for this class.

By using the q-derivative operator and the Legendre polynomials, some new subclasses of q-starlike functions and bi-univalent functions are introduced. Several coefficient estimates and Fekete–Szegö-type inequalities for functions in each of these subclasses are obtained. The results derived in this article are shown to extend and generalize those in some earlier works. [ABSTRACT FROM AUTHOR]

In the article we introduce and investigate several new subclasses of q-starlike and q-convex type analytic and multivalent functions involving a generalized Bernardi integral operator, and establish the Fekete-Szegö type functional inequalities for these function classes. Besides, the corresponding bound estimates of the coefficients ap+1 and ap+2 are obtained.

In this paper, we first investigate some subclasses of q-starlike functions. We then apply higher-order q-derivative operators to introduce and study a new subclass of q-starlike functions, which involves the Janowski functions. Several coefficient inequalities and a sufficient condition are derived. Relevant connections with a number of earlier works on this subject are also pointed out.

By utilizing the concept of the q-fractional derivative operator and bi-close-to-convex functions, we define a new subclass of A, where the class A contains normalized analytic functions in the open unit disk E and is invariant or symmetric under rotation. First, using the Faber polynomial expansion (FPE) technique, we determine the lth coefficient bound for the functions contained within this class. We provide a further explanation for the first few coefficients of bi-close-to-convex functions defined by the q-fractional derivative. We also emphasize upon a few well-known outcomes of the major findings in this article.

In this article, a new linear extended multiplier operator is defined utilizing the q-Choi–Saigo–Srivastava operator and the q-derivative. Two generalized subclasses of q—uniformly convex and starlike functions of order δ—are defined and studied using this new operator. Necessary conditions are derived for functions to belong in each of the two subclasses, and subordination theorems involving the Hadamard product of such particular functions are stated and proven. As applications of those findings using specific values for the parameters of the new subclasses, associated corollaries are provided. Additionally, examples are created to demonstrate the conclusions’ applicability in relation to the functions from the newly introduced subclasses.

In this paper, we introduce and investigate two new subclasses of analytic and bi-univalent functions using the q-derivative operator Dq0

The potential for widespread applications of the geometric and mapping properties of functions of a complex variable has motivated this article. On the other hand, the basic or quantum (or q-) derivatives and the basic or quantum (or q-) integrals are extensively applied in many different areas of the mathematical, physical and engineering sciences. Here, in this article, we first apply the q-calculus in order to introduce the q-derivative operator Sη,p,qn,m. Secondly, by means of this q-derivative operator, we define an interesting subclass Tℵλ,pn,m(η,α,κ) of the class of normalized analytic and multivalent (or p-valent) functions in the open unit disk U. This p-valent analytic function class is associated with the class κ-UCV of κ-uniformly convex functions and the class κ-UST of κ-uniformly starlike functions in U. For functions belonging to the normalized analytic and multivalent (or p-valent) function class Tℵλ,pn,m(η,α,κ), we then investigate such properties as those involving (for example) the coefficient bounds, distortion results, convex linear combinations, and the radii of starlikeness, convexity and close-to-convexity. We also consider a number of corollaries and consequences of the main findings, which we derived herein.

The main object of this paper is to investigate convolution properties and coefficient estimates for some subclasses of multivalent functions defined by q-derivative operator in the open unit disc. The results presented here would provide extensions of those given in earlier works. [ABSTRACT FROM AUTHOR]

In this article, we define a q-derivative operator of univalent functions associated with the q-exponential function. Moreover, we introduce differential subordination and differential superordination for the subordination class defined by this operator. Sandwich-type theorems of several known results also are derived by applying these results. [ABSTRACT FROM AUTHOR]

In this paper our aim is to study some valuable problems dealing with newly defined subclass of multivalent $q$-starlike functions. These problems include the initial coefficient estimates, Toeplitz matrices, Hankel determinant, Fekete-Szego problem, upper bounds of the functional $\left \vert a_{p+1}-\mu a_{p+1}^{2}\right \vert $ for the subclass of multivalent $q$-starlike functions. As applications we study a $% q $-Bernardi integral operator for a subclass of multivalent $q$-starlike functions. Furthermore, we also highlight some known consequence of our main results.

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.

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.

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.

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.

In the present investigation we introduce two new subclasses M S ∗ q (p, b; c) and M K q(p, b; c) of meromorphic multivalent functions by using q-derivative operator defined in the punctured unit disc. We use the Cassinian Oval √ 1 + cz with c ∈ (0, 1] as a subordinant function in this treatise. Also we derive several properties including convolution properties, the necessary and sufficient condition and coefficient estimates for these subclasses. [ABSTRACT FROM AUTHOR]

n the present paper, two subclasses MSpq(b; A;B) and MKp q(b; A;B) of meromorphic multivalent functions by using q-derivative operator are defined in the punctured unit disc. Also, several properties including convolution properties, the necessary and suficient condition and coeficient estimates for these subclasses are derived. [ABSTRACT FROM AUTHOR]

Our goal is to study the addition of a Dirac mass to a Hq-Laguerre-Hahn form where Hq be the q-derivative operator. The Hq-Laguerre-Hahn character and the class of the obtained form is discussed into detail. An example in connection with the first order associated of a Hq-classical form is highlighted. [ABSTRACT FROM AUTHOR]

Abstract A number of families of q-extensions of analytic functions in the open unit disk U $\mathbb{U}$ have been defined by means of basic (or q-)calculus and considered from many distinctive prospectives and viewpoints. In this paper, we generalize and study certain subclasses of analytic functions involving higher-order q-derivative operators. We settle characteristic equations for these presumably new classes and also study numerous coefficient inequalities. For the results obtained in this presentation, we also carry out appropriate connections with those in multiple other concerning works on this subject.

In this paper, the concepts of quantum (or q-) calculus and conic regions are combined to define a new domain Ωk,q,γ which represents the generalized conic regions. Then by using a certain generalized conic domain Ωk,q,γ we define and investigate a new subclass of normalized analytic functions in open unit disk E. We also investigate a number of useful properties and characteristics of this subclass such as, structural formula, necessary and sufficient condition, coefficient estimates, Feketo-Szego problem, distortion inequalities, closure theorem, and subordination result. We also highlight some known consequences of our main results as corollaries.

This research presents a new group of mathematical functions connected to Bernoulli's Lemniscate, using the q -derivative. Expanding on previous studies, the research concentrates on determining coefficient approximations, the Fekete-Szego functional, Zalcman inequality, Krushkal inequality, along with the second and third Hankel determinants for this recently established collection of functions. Additionally, the study derives the Fekete-Szego inequality for the function ξ f ( ξ ) and obtains the inverse function f - 1 ( ξ ) for this specific class. This research advances our understanding in this area and suggests for further exploration., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Authors.)

This paper considers the basic concepts of q-calculus and the principle of subordination. We define a new subclass of q-starlike functions related to the Salagean q-differential operator. For this class, we investigate initial coefficient estimates, Hankel determinants, Toeplitz matrices, and Fekete-Szegö problem. Moreover, we consider the q-Bernardi integral operator to discuss some applications in the form of some results.

By using the q-derivative operator and the Legendre polynomials, some new subclasses of q-starlike functions and bi-univalent functions are introduced. Several coefficient estimates and Fekete–Szegö-type inequalities for functions in each of these subclasses are obtained. The results derived in this article are shown to extend and generalize those in some earlier works.