By using a certain general conic domain as well as the quantum (or q-) calculus, here we define and investigate a new subclass of normalized analytic and starlike functions in the open unit disk $\mathbb{U}$ . In particular, we find the Hankel determinant and the Toeplitz matrices for this newly-defined class of analytic q-starlike functions. We also highlight some known consequences of our main results.

By making use of the concept of basic (or q-) calculus, various families of q-extensions of starlike functions of order $\alpha$ in the open unit disk $\mathbb{U}$ were introduced and studied from many different viewpoints and perspectives. In this paper, we first investigate the relationship between various known classes of q-starlike functions that are associated with the Janowski functions. We then introduce and study a new subclass of q-starlike functions that involves the Janowski functions. We also derive several properties of such families of q-starlike functions with negative coefficients including (for example) distortion theorems.

The main purpose of this article is to find the upper bound of the third Hankel determinant for a family of q-starlike functions which are associated with the Ruscheweyh-type q-derivative operator. The work is motivated by several special cases and consequences of our main results, which are pointed out herein.

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 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.