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This paper introduces a new subclass of harmonic functions with a positive real part, denoted by $ HP_q(\beta) $, where $ 0 \leq \beta < 1 $ and $ 0 < q < 1 $. A sufficient coefficient condition is established for functions within this class, which is also necessary when dealing with negative coefficients. In addition, the growth theorem is derived, and the extreme points associated with this subclass are also identified. Finally, the $ q $-integral operator for harmonic functions of the form $ f = h + g $ with a positive real part is presented.

In this paper, we introduced and studied two new classes of analytic functions using the concepts of subordination and q-calculus. We established inclusion relations for these q-classes and integral-preserving properties associated with the q-integral operator. We also determined certain convolution properties.

In the present research work, we have derived two theorems which involves integral operators of Erd'elyi-Kober type and a q-analogue of modified multivariable I-function. The related averment for the Riemann-Liouville and Weyl fractional basic integral transforms are also deduced. A number of corollaries concerning the basic analogue of modified multivariable H-function, q - analogue of multivariable H-function and remarks are given at the end of this paper. [ABSTRACT FROM AUTHOR]

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.

In the present investigation, our aim is to define a generalized subclass of analytic and bi-univalent functions associated with a certain $q$-integral operator in the open unit disk $\mathbb{U}$. We estimate bounds on the initial Taylor-Maclaurin coefficients $\left \vert a_{2}\right \vert$ and $\left \vert a_{3}\right \vert $ for normalized analytic functions $f$ in the open unit disk by considering the function $f$ and its inverse $g = f^{{-}{1}}$. Furthermore, we derive special consequences of the results presented here, which would apply to several (known or new) subclasses of analytic and bi-univalent 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.

In this paper, we introduce a generalization of the Kantorovich-type Bernstein operators based on q-integers and get a Bohman-Korovkin-type approximation theorem of these operators. We also compute the rate of convergence using the first modulus of smoothness. [ABSTRACT FROM AUTHOR]

In this paper, we solve dual and triple sequences involvingq-orthogonal polynomials. We also introduce and solve a system of dual series equations when the kernel is theq-Laguerre polynomials. Examples are included. [ABSTRACT FROM PUBLISHER]