A Subclass of Multivalent Janowski Type q-Starlike Functions and Its Consequences

Abstract

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 Ap, where class Ap 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 (p,q)-variations, by making some straightforward simplification and will be an inconsequential exercise simply because the additional parameter p is obviously unnecessary.

A Subclass of Janowski Starlike Functions Involving Mathieu-Type Series

“…Furthermore, this functions class has been generalized and studied by the many authors. For example, recently in [12], Hu et al defined a new subclass of multivalent Janowski functions and found out some of its interesting properties. In particular, they made use of certain basic (or q-) calculus in order to define their class.…”

“…Then, they gave certain interesting results, like coefficient bounds, radii of starlikeness and convexity, sufficiency criteria, growth theorem and distortion problem. In their paper published in Symmetry (see [12]), open some interesting step toward a more aggregate and comprehensive analysis of these functions. In our present work, we are essentially motivated by the work of Hu et al [12].…”

“…In their paper published in Symmetry (see [12]), open some interesting step toward a more aggregate and comprehensive analysis of these functions. In our present work, we are essentially motivated by the work of Hu et al [12]. In particular, we make use of certain Mathieu-type series and the Janowski functions in order to define our functions class.…”

Certain New Subclass of Multivalent Q-Starlike Functions Associated with Q-Symmetric Calculus

“…In [10,11], Arif et al defined and investigated some new subclasses of multivalent functions by implementing the concept of the q-derivative operator, while Zang et al [12] used the basic concepts of q-calculus to define the generalized conic domain. For some recent investigations of q-function theory, we refer readers to [13][14][15][16][17][18][19].…”

Coefficient Estimates for a Subclass of Meromorphic Multivalent q-Close-to-Convex Functions

“…In this article, we are essentially motivated by the recently published paper of Hu et al in Symmetry (see [26]) and some other related works as discussed above (see for example [27][28][29][30][31]), we now define a subclass MK µ,q (p, m, A, B) of A p by using the operator D m µ,q as follows.…”