Q-convex Functions Research Articles

Certain New Applications of Symmetric q-Calculus for New Subclasses of Multivalent Functions Associated with the Cardioid Domain

In this work, we study some new applications of symmetric quantum calculus in the field of Geometric Function Theory. We use the cardioid domain and the symmetric quantum difference operator to generate new classes of multivalent q-starlike and q-convex functions. We examine a wide range of interesting properties for functions that can be classified into these newly defined classes, such as estimates for the bounds for the first two coefficients, Fekete–Szego-type functional and coefficient inequalities. All the results found in this research are sharp. A number of well-known corollaries are additionally taken into consideration to show how the findings of this research relate to those of earlier studies.

Coefficient Inequalities for q-Convex Functions with Respect to q-Analogue of the Exponential Function

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.

Hartogs’ analyticity theorem for C2-mappings and maximum principle for q-convex functions

Hartogs’ analyticity theorem for C2-mappings and maximum principle for q-convex functions

Applications of Fractional Differential Operator to Subclasses of Uniformly q-Starlike Functions

In this paper, we use the concept of quantum (or q-) calculus and define a q-analogous of a fractional differential operator and discuss some of its applications. We consider this operator to define new subclasses of uniformly q-starlike and q-convex functions associated with a new generalized conic domain, Λβ,q,γ. To begin establishing our key conclusions, we explore several novel lemmas. Furthermore, we employ these lemmas to explore some important features of these two classes, for example, inclusion relations, coefficient bounds, Fekete–Szego problem, and subordination results. We also highlight many known and brand-new specific corollaries of our findings.

New Applications of the Sălăgean Quantum Differential Operator for New Subclasses of q-Starlike and q-Convex Functions Associated with the Cardioid Domain

In this paper, we define a new family of q-starlike and q-convex functions related to the cardioid domain utilizing the ideas of subordination and the Sălăgean quantum differential operator. The primary contribution of this article is the derivation of a sharp inequality for the newly established subclasses of q-starlike and q-convex functions in the open unit disc U. For this novel family, bounds of the first two Taylor–Maclaurin coefficients, the Fekete–Szegö-type functional, and coefficient inequalities are studied. Furthermore, we also investigate some new results for the inverse function belonging to the classes of q-starlike and q-convex functions. The results presented in this article are sharp. To draw connections between the early and present findings, several well-known corollaries are also highlighted. Symmetric quantum calculus operator theory can be used to investigate the symmetry properties of this new family of functions.

Some New Applications of the q-Analogous of Differential and Integral Operators for New Subclasses of q-Starlike and q-Convex Functions

In the geometric function theory of complex analysis, the investigation of the geometric properties of analytic functions using q-analogues of differential and integral operators is an important area of study, offering powerful tools for applications in numerical analysis and the solution of differential equations. Many topics, including complex analysis, hypergeometric series, and particle physics, have been generalized in q-calculus. In this study, first of all, we define the q-analogues of a differential operator (DRλ,qm,n) by using the basic idea of q-calculus and the definition of convolution. Additionally, using the newly constructed operator (DRλ,qm,n), we establish the q-analogues of two new integral operators (Fλ,γ1,γ2,…γlm,n,q and Gλ,γ1,γ2,…γlm,n,q), and by employing these operators, new subclasses of the q-starlike and q-convex functions are defined. Sufficient conditions for the functions (f) that belong to the newly defined classes are investigated. Additionally, certain subordination findings for the differential operator (DRλ,qm,n) and novel geometric characteristics of the q-analogues of the integral operators in these classes are also obtained. Our results are generalizations of results that were previously proven in the literature.

Studying the Harmonic Functions Associated with Quantum Calculus

Using the derivative operators’ q-analogs values, a wide variety of holomorphic function subclasses, q-starlike, and q-convex functions have been researched and examined. With the aid of fundamental ideas from the theory of q-calculus operators, we describe new q-operators of harmonic function Hϱ,χ;qγF(ϖ) in this work. We also define a new harmonic function subclass related to the Janowski and q-analog of Le Roy-type functions Mittag–Leffler functions. Several important properties are assigned to the new class, including necessary and sufficient conditions, the covering Theorem, extreme points, distortion bounds, convolution, and convex combinations. Furthermore, we emphasize several established remarks for confirming our primary findings presented in this study, as well as some applications of this study in the form of specific outcomes and corollaries.

Correction to: Some subclasses of p-valent $$\gamma $$-uniformly type q-starlike and q-convex functions defined by using a certain generalized q-Bernardi integral operator

Correction to: Some subclasses of p-valent $$\gamma $$-uniformly type q-starlike and q-convex functions defined by using a certain generalized q-Bernardi integral operator

A new q-fractional integral operator and its applications to the coefficient problem involving the second Hankel determinant for q-starlike and q-convex functions

A new q-fractional integral operator and its applications to the coefficient problem involving the second Hankel determinant for q-starlike and q-convex functions

Some subclasses of p-valent $$\gamma $$-uniformly type q-starlike and q-convex functions defined by using a certain generalized q-Bernardi integral operator

Some subclasses of p-valent $$\gamma $$-uniformly type q-starlike and q-convex functions defined by using a certain generalized q-Bernardi integral operator

Applications of Higher-Order q-Derivative to Meromorphic q-Starlike Function Related to Janowski Function

By making use of a higher-order q-derivative operator, certain families of meromorphic q-starlike functions and meromorphic q-convex functions are introduced and studied. Several sufficient conditions and coefficient inequalities for functions in these subclasses are derived. The results presented in this article extend and generalize a number of previous results.

Certain subclasses of meromorphic multivalent q-starlike and q-convex functions

Abstract In this paper, we extend the idea of q-derivative (or q-difference) operator for meromorphic p-valent functions and introduce certain new subclasses of meromorphic multivalent q-starlike and q-convex functions. We investigate subordination properties, coefficient inequalities, convolution properties and integral representations. We also derive sufficient conditions for meromorphic multivalent q-starlike and q-convex functions and highlight some known consequence of our main results.

Sufficient condition for q-starlike and q-convex functions associated with normalized gauss hypergeometric function

The main object of this paper is to investigate and determine a sufficient condition for q-starlikeness and q-convexity for functions which are associated with normalized Gauss hypergeometric function.

A conjecture on Marx-Strohhäcker type inclusion relation between q-convex and q-starlike functions

We prove that the class K of normalized univalent convex functions defined in the unit disk E is contained in Kq(1−q1+q2)(0<q<1), the class of normalized univalent q-convex functions of order (1−q)/(1+q2). We provide examples that exhibit a Marx-Strohhäcker type inclusion relation, i.e. Kq(1−q1+q2)⊂Sq⁎(11+q), where Sq⁎(11+q) is the class of q-starlike functions of order 1/(1+q). Note that for q→1− this relation coincides with the well-known result, K⊂S⁎(12), of Marx and Strohhäcker.

Suficient condition for q-starlike and q-convex functions associated with generalized confluent hypergeometric function

The main object of this paper is to investigate and determine a sufficient condition for q-starlike and q-convex functions which are associated with generalized confluent hypergeometric function. The main object of this paper is to investigate and determine a sufficient condition for q-starlike and q-convex functions which are associated with generalized confluent hypergeometric function.The main object of this paper is to investigate and determine a sufficient condition for q-starlike and q-convex functions which are associated with generalized confluent hypergeometric function.

Certain Subclasses of β-Uniformly q-Starlike and β-Uniformly q-Convex Functions

In this paper, the authors introduced certain subclasses β-uniformly q-starlike and β-uniformly q-convex functions of order α involving the q-derivative operator defined in the open unit disc. Coefficient bounds were also investigated.

Fekete Szegö properties for a class of univalent functions of complex order associated with q-difference operator

In this paper we introduce new class of q-starlike and q-convex functions of complex order involving the q-derivative operator. Morever, we find estimates on the coefficients for functions in this class.

Coefficient Inequalities of q-starlike and q-convex Functions Defined by q-Derivative

Coefficient Inequalities of q-starlike and q-convex Functions Defined by q-Derivative

Coefficient estimates for the class of quasi q-convex functions

In this paper we introduce and investigate the class of $ P_{q}(\lambda,\beta, A, B)$, which is called quasi q-starlike and quasi q-convex with respect to the values of the parameter $\lambda$. We give coefficient bounds estimates and the results for the main theorem.

The Fekete–Szegö coefficient functional problems for q-starlike and q-convex functions related with lemniscate of Bernoulli

The area of [Formula: see text]-calculus has attracted the serious attention of researchers. This great interest is due its application in various branches of mathematics and physics. The application of [Formula: see text]-calculus was initiated by Jackson [Jackson, On [Formula: see text]-definite integrals, Quart. J. Pure Appl. Math. 41 (1910) 193–203; On [Formula: see text]-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh 46 (1908) 253–281.], who was the first to develop [Formula: see text]-integral and [Formula: see text]-derivative in a systematic way. In this paper, we make use of the concept of [Formula: see text]-calculus in the theory of univalent functions, to obtain the bounds for certain coefficient functional problems of [Formula: see text]-starlike and [Formula: see text]-convex functions. Further, we also obtain similar type of inequalities related to lemniscate of Bernoulli. The authors sincerely hope that this paper will revive this concept and encourage other researchers to work in this [Formula: see text]-calculus in the near-future in the area of complex function theory. Also, we present a direct and shortened proof for the estimates of [Formula: see text] found in [Mishra and Gochhayat, Fekete–Szego problem for [Formula: see text]-uniformly convex functions and for a class defined by Owa–Srivastava operator, J. Math. Anal. Appl. 347(2) (2008) 563–572] for [Formula: see text], [Formula: see text].