A Note on Certain Classes of Close-to-Convex Functions

Al-Kharsani, H. A.

doi:10.5556/j.tkjm.30.1999.4223

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“…Indeed, for different choices of the parameters α, κ and p, we obtain many new or known subclasses, some of which were studied in the earlier literature (see, for example, [7][8][9][10][17][18][19]).…”

Section: Introduction and Definitionsmentioning

confidence: 99%

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

2023

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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 Sη,p,qn,m. Secondly, by means of this q-derivative operator, we define an interesting subclass Tℵλ,pn,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 Tℵλ,pn,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.

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