Citation: Hu, Q.; Badar, R.S.; Khan, M.G. Subclasses of q-Uniformly Starlike Functions Obtained via the q-Carlson–Shaffer Operator. Axioms 2024, 13, 842. https://doi.org/ 10.3390/axioms13120842 Academic Editor: Georgia Irina Oros Received: 29 October 2024 Revised: 26 November 2024 Accepted: 28 November 2024 Published: 29 November 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Article Subclassesof q-Uniformly Starlike Functions Obtained via the q-Carlson–Shaffer Operator Qiuxia Hu 1 , Rizwan Salim Badar 2 and Muhammad Ghaffar Khan 3, * 1 Department of Mathematics, Luoyang Normal University, Luoyang 471934, China; huqiuxia306@163.com 2 Department of Mathematics, Allama Iqbal Open University, Islamabad 44000, Pakistan; rizwan.salim@aiou.edu.pk 3 Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat 26000, Pakistan * Correspondence: ghaffarkhan020@gmail.com Abstract: This article investigates the applications of the q-Carlson–Shaffer operator on subclasses of q-uniformly starlike functions, introducing the class ST q (m, c, d, β). The study establishes a necessary condition for membership in this class and examines its behavior within conic domains. The article delves into properties such as coefficient bounds, the Fekete–Szegö inequality, and criteria defined via the Hadamard product, providing both necessary and sufficient conditions for these properties. Keywords: analytic function; conic domain; quantum calculus; q-Carlson–Shaffer operator MSC: 30C45; 30C50; 30C80; 33D15 1. Introduction Quantum calculus, sometimes called q-calculus, is a branch of mathematics that gener- alizes traditional calculus by introducing a parameter q to replace limits in differentiation and integration. In quantum calculus, instead of approaching infinitesimal changes, the calculus operates over discrete steps, where the parameter q controls the granularity of these steps. This approach is particularly useful in areas that naturally involve discrete structures, such as quantum theory, number theory, and combinatorics. The concepts of derivative and integral within the framework of q-calculus were formally introduced by Jackson—see [1,2]—establishing the foundational principles of q-calculus. For a more comprehensive understanding, please consult [3]. q-Calculus has extensive applications across various fields of mathematics and physics, including the calculus of variations, quantum groups, the theory of relativity, combinatorics, and quantum mechanics. This wide-ranging applicability underscores the importance of q-calculus as a formidable math- ematical instrument. Recently, the use of q-calculus in Geometric Function Theory (GFT) has attracted considerable attention from scholars. A significant application is presented in [4], where Srivastava employed q-hypergeometric functions within the context of GFT. Furthermore, the modified q-derivative has been utilized to define q-starlike functions, as outlined in [5,6]. The exploration of the q-generalization of close-to-convex functions is discussed in [7], while [8,9] introduce generalized forms of q-starlike and q-close-to-convex functions, leading to the development of modified subclasses of analytic functions within the q-calculus framework. The categorization of q-convex functions with a complex order, which generalizes several traditional classes in GFT, is presented in [10]. Additionally, the q-extensions of starlike functions, particularly those associated with Janowski func- tions, are established in [11]. In the realm of GFT, linear operators have been constructed and utilized to characterize new subclasses of analytic functions, resulting in significant progress, especially with the recent introduction of q-special functions defined in terms of the q-Pochhammer symbol, q-shifted factorial, q-Gamma function, q-hypergeometric function, and deformed q-Lerch–Hurwitz function. The advancement of q-convoluted Axioms 2024, 13, 842. https://doi.org/10.3390/axioms13120842 https://www.mdpi.com/journal/axioms