Some New Applications of the <i>q</i>-Analogous of Differential and Integral Operators for New Subclasses of <i>q</i>-Starlike and <i>q</i>-Convex Functions

Abstract

In the geometric function theory of complex analysis, the investigation of the geometric properties of analytic functions using q-analogues of differential and integral operators is an important area of study, offering powerful tools for applications in numerical analysis and the solution of differential equations. Many topics, including complex analysis, hypergeometric series, and particle physics, have been generalized in q-calculus. In this study, first of all, we define the q-analogues of a differential operator (DRλ,qm,n) by using the basic idea of q-calculus and the definition of convolution. Additionally, using the newly constructed operator (DRλ,qm,n), we establish the q-analogues of two new integral operators (Fλ,γ1,γ2,…γlm,n,q and Gλ,γ1,γ2,…γlm,n,q), and by employing these operators, new subclasses of the q-starlike and q-convex functions are defined. Sufficient conditions for the functions (f) that belong to the newly defined classes are investigated. Additionally, certain subordination findings for the differential operator (DRλ,qm,n) and novel geometric characteristics of the q-analogues of the integral operators in these classes are also obtained. Our results are generalizations of results that were previously proven in the literature.

Keywords

• analytic functions
• convolution
• quantum (or <i>q</i>-) calculus
• <i>q</i>-difference operator
• <i>q</i>-integral operator
• <i>q</i>-starlike and <i>q</i>-convex functions