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.

The Faber polynomial expansion method and its application to the general coefficient problem for some subclasses of bi-univalent functions associated with a certain $q$-integral operator

In this paper, we use q-derivative operator to define a new class of q-starlike functions associated with k-Fibonacci numbers. This newly defined class is a subclass of class $\mathcal{A}$ of normalized analytic functions, where class $\mathcal{A}$ is invariant (or symmetric) under rotations. For this function class we obtain an upper bound of the third Hankel determinant.

In the present investigation, by using certain higher-order q-derivatives, the authors introduce and investigate several new subclasses of the family of multivalent q-starlike functions in the open unit disk. For each of these newly-defined function classes, several interesting properties and characteristics are systematically derived. These properties and characteristics include (for example) distortion theorems and radius problems. A number of coefficient inequalities and a sufficient condition for functions belonging to the subclasses studied here are also discussed. Relevant connections of the various results presented in this investigation with those in earlier works on this subject are also pointed out.