In this paper, the concepts of quantum (or q-) calculus and conic regions are combined to define a new domain Ω_{k, q, γ} which represents the generalized conic regions. Then by using a certain generalized conic domain Ω_{k, 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.

We introduce in our present investigation a new subclass of analytic and biunivalent functions associated with Ruscheweyh q-differential operator in open unit disk E. We use the Faber polynomial expansions to find nth coefficients bounds of class of bisubordinate functions and also find initial coefficient estimates.

Certain new classes of q-convex and q-close to convex functions that involve the q-Janowski type functions have been defined by using the concepts of quantum (or q-) calculus as well as q-conic domain $\left({\Omega }_{k,q}[\lambda ,\alpha ]\right)$ . This study explores some important geometric properties such as coefficient estimates, sufficiency criteria and convolution properties of these classes. A distinction of new findings with those obtained in earlier investigations is also provided, where appropriate.

In our present study, we introduce and examine several new subclasses of analytic functions defined through the convolution operator $H_Y^{\lambda ,q}$, which is formulated using the classical error function. Our approach relies on the concept of quasi-subordination, a broad extension of subordination in geometric function theory. For each of these newly defined classes, we focus on deriving significant properties such as the upper bounds of the first few Taylor–Maclaurin coefficients of normalized series, evaluation of classical Fekete–Szegö functional, and calculating the upper bounds of Hankel and Toeplitz determinants for different orders. The combination of the proposed operator and the quasi-subordination framework offers a unified strategy for tackling these problems. Our findings also generalize various existing results in the field.

In our present investigation, we extend the work of [1] and define a new subclass of analytic functions in conic regions using the linear -difference operator. In [1], certain new classes of analytic functions were introduced, and their properties were widely investigated. They derived various subordination conditions and coefficient bounds. In this article, we obtain several coefficient bounds and other subordination results for functions belonging to the newly defined class. These coefficient bounds can be used to estimate the area and arc length problems in Geometric Function Theory.