Exploring the Geometric Characteristics of Harmonic Functions Using the Second-Order Jackson q-Derivative

Abstract

In this paper, we introduce a new subclass of harmonic functions defined by Jackson’s second-order \(q\)-derivative. This subclass, denoted as \(\mathcal{AH}_q(\sigma, \tau, \rho)\), is characterized by a higher-order differential inequality involving both the first and second \(q\)-derivatives. We establish the necessary and sufficient conditions for a function to belong to this class and investigate its geometric properties, such as coefficient bounds, distortion bounds, and closure under convolution. Additionally, we analyze the relationship between \(\mathcal{AH}_q(\sigma, \tau, \rho)\) and a related subclass of analytic functions, denoted as \(\mathcal{A}_q(\sigma, \tau, \rho))\). Furthermore, we explore the radii of convexity and starlikeness for the functions in these classes, providing significant contributions to the field of geometric function theory. Our results generalize and extend existing works on harmonic mappings defined by higher-order differential operators.