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.

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.

First, by making use of the concept of basic (or q-) calculus, as well as the principle of subordination between analytic functions, generalization ${\mathcal{R}}_q\left(h\right)$ of the class $\mathcal{R}\left(h\right)$ of analytic functions, which are associated with the leaf-like domain in the open unit disk $\mathbb{U}$, is given. Then, the coefficient estimates, the Fekete–Szegö problem, and the second-order Hankel determinant $H_2\left(1\right)$ for functions belonging to this class ${\mathcal{R}}_q\left(h\right)$ are investigated. Furthermore, similar results are examined and presented for the functions $\frac{z}{f\left(z\right)}$ and $f^{-1}\left(z\right)$. For the validity of our results, relevant connections with those in earlier works are also pointed out.

In our present investigation, with the help of the basic (or q-) calculus, we first define a new domain which involves the Janowski function. We also define a new subclass of the class of q-starlike functions, which maps the open unit disk $\mathbb{U}$, given by $\mathbb{U}= \left\{z:z\in \mathbb{C} \mathrm{and} \left|z\right| <1\right\},$ onto this generalized conic type domain. We study here some such potentially useful results as, for example, the sufficient conditions, closure results, the Fekete-Szegö type inequalities and distortion theorems. We also obtain the lower bounds for the ratio of some functions which belong to this newly-defined function class and for the sequences of the partial sums. Our results are shown to be connected with several earlier works related to the field of our present investigation. Finally, in the concluding section, we have chosen to reiterate the well-demonstrated fact that any attempt to produce the rather straightforward $(p,q)$-variations of the results, which we have presented in this article, will be a rather trivial and inconsequential exercise, simply because the additional parameter p is obviously redundant.

A number of families of q-extensions of analytic functions in the open unit disk \(\mathbb{U}\) have been defined by means of basic (or q-)calculus and considered from many distinctive prospectives and viewpoints. In this paper, we generalize and study certain subclasses of analytic functions involving higher-order q-derivative operators. We settle characteristic equations for these presumably new classes and also study numerous coefficient inequalities. For the results obtained in this presentation, we also carry out appropriate connections with those in multiple other concerning works on this subject.