Four-Leaf Inequalities: A q-Series Approach

Abstract

This research delves into several inequalities of complex functions, which are key to understanding their behavior. Such inequalities are vital in complex analysis for setting bounds, demonstrating convergence, and uncovering analytical and geometrical characteristics of regular q-functions. Our investigation focuses on a new collection of regular and univalent q-functions, denoted by  and defined within the unit disk: |z| < 1. The definition and study of functions in , encompass several established subsets of regular q-functions, including q-bounded turning functions, Carathéodory functions, and four-leaf-type function. The study also incorporates several foundational mathematical ideas: a fresh Mathieu q-series expansion, subordination and convolution principles, q-operator theory, transformation of complex functions, and a few domain mappings. In particular, we explore the properties of two regular functions of the forms:  _. Here, f is the jth-root transform of g. Our findings include estimates for some prominent inequalities such as Fekete-Szegö, Zalcman, Krushkal, and Ma inequalities. We also include numerous corollaries to highlight the connection between g and f.