Subclasses of p -Valent κ -Uniformly Convex and Starlike Functions Defined by the q -Derivative Operator

The potential for widespread applications of the geometric and mapping properties of functions of a complex variable has motivated this article. On the other hand, the basic or quantum (or q -) derivatives and the basic or quantum (or q -) integrals are extensively applied in many different areas of the mathematical, physical and engineering sciences. Here, in this article, we first apply the q -calculus in order to introduce the q -derivative operator S η , p , q n , m . Secondly, by means of this q -derivative operator, we define an interesting subclass T ℵ λ , p n , m ( η , α , κ ) of the class of normalized analytic and multivalent (or p -valent) functions in the open unit disk U . This p -valent analytic function class is associated with the class κ - UCV of κ -uniformly convex functions and the class κ - UST of κ -uniformly starlike functions in U . For functions belonging to the normalized analytic and multivalent (or p -valent) function class T ℵ λ , p n , m ( η , α , κ ) , we then investigate such properties as those involving (for example) the coefficient bounds, distortion results, convex linear combinations, and the radii of starlikeness, convexity and close-to-convexity. We also consider a number of corollaries and consequences of the main findings, which we derived herein.