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

Qiuxia Hu,Bakhtiar Ahmad,Nazar Khan,Hari M Srivastava,Muhammad Ghaffar Khan,Wali Khan Mashwani,Bilal Khan

doi:10.3390/sym13071275

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.

Highlights

Srivastava et al [11,12] first defined certain subclasses of q-starlike functions and studied their various properties including for example some coefficient inequalities, inclusion properties, and a number of sufficient conditions
Before starting radii problems let us remaind the definition of important classes of multivalent starlike and convex functions
By Cp(σ) we mean the class of multivalent convex functions, that is a function f ∈ Ap and satisfies the inequality below p2 f (z)

Summary

Introduction

Srivastava et al [11,12] first defined certain subclasses of q-starlike functions and studied their various properties including for example some coefficient inequalities, inclusion properties, and a number of sufficient conditions. Let Ap be the class of analytic and multivalent (or p-valent) functions f (z) in the open unit disk A function f ∈ Ap is said to be in the class Sq(p, α, A, B), if it satisfies the following subordination condition: 1 1−α zDq [p]q f (z) f (z)

Results

Conclusion