In recent years, the usage of the q‐derivative and symmetric q‐derivative operators is significant. In this study, firstly, many known concepts of the q‐derivative operator are highlighted and given. We then use the symmetric q‐derivative operator and certain q‐Chebyshev polynomials to define a new subclass of analytic and bi‐univalent functions. For this newly defined functions’ classes, a number of coefficient bounds, along with the Fekete–Szegö inequalities, are also given. To validate our results, we give some known consequences in form of remarks.

By making use of the concept of basic (or q-) calculus, many subclasses of analytic and symmetric q-starlike functions have been defined and studied from different viewpoints and perspectives. In this article, we introduce a new class of meromorphic multivalent close-to-convex functions with the help of a q-differential operator. Furthermore, we investigate some useful properties such as sufficiency criteria, coefficient estimates, distortion theorem, growth theorem, radius of starlikeness, and radius of convexity for this new subclass.

In this article, by utilizing the theory of quantum (or q-) calculus, we define a new subclass of analytic and multivalent (or p-valent) functions class ${\mathcal{A}}_p$, where class ${\mathcal{A}}_p$ is invariant (or symmetric) under rotations. The well-known class of Janowski functions are used with the help of the principle of subordination between analytic functions in order to define this subclass of analytic and p-valent functions. This function class generalizes various other subclasses of analytic functions, not only in classical Geometric Function Theory setting, but also some q-analogue of analytic multivalent function classes. We study and investigate some interesting properties such as sufficiency criteria, coefficient bounds, distortion problem, growth theorem, radii of starlikeness and convexity for this newly-defined class. Other properties such as those involving convex combination are also discussed for these functions. In the concluding part of the article, we have finally given the well-demonstrated fact that the results presented in this article can be obtained for the $(\mathfrak{p},q)$-variations, by making some straightforward simplification and will be an inconsequential exercise simply because the additional parameter $\mathfrak{p}$ is obviously unnecessary.

In our present investigation, some coefficient functionals for a subclass relating to starlike functions connected with three-leaf mappings were considered. Sharp coefficient estimates for the first four initial coefficients of the functions of this class are addressed. Furthermore, we obtain the Fekete–Szegö inequality, sharp upper bounds for second and third Hankel determinants, bounds for logarithmic coefficients, and third-order Hankel determinants for two-fold and three-fold symmetric functions.

In this article we define a class of starlike functions with respect to symmetric points in the domain of sine function. Also, we investigate coefficients bounds and upper bounds for the third order Hankel determinant for this defined class. We also evaluate the Zalcman functional \(|a_{3}^{2}-a_{5}|\). Specializing the parameters, we improve Zalcman functional for the class of starlike functions.

In analysis, the introduction of q-calculus has been a revelation. It has a deep impact on various concepts and applications of pure and applied sciences. In this article we investigate certain geometric properties relating to convolution of functions of a newly defined class of analytic functions. The important region of the lemniscate of Bernoulli is considered. Here we utilize concepts of q-calculus which enhances and generalizes the vitality of this research work. In the same context we study the Fekete–Szegö problem.