Geometric Properties and Applications in System Modeling for a Generalized q-Symmetric Operator

This paper introduces a novel generalized q-symmetric differential operator for studying a certain subclass of univalent functions with negative coefficients. We establish several significant theoretical results for this class, including sharp coefficient bounds and characterization theorems based on the generalized Hadamard product. Two significant applications demonstrate the theoretical framework’s practical utility. First, in the context of geometric modeling, we demonstrate how the function class and operator can be utilized to create and control complex, non-overlapping transformations. Second, in digital signal processing, we show that these functions serve as stable digital filter prototypes and that our operator is an effective tool for fine-tuning the filter’s frequency response. These applications bridge the gap between abstract geometric function theory and practical system design by demonstrating the operator’s versatility as a tool for analysis and synthesis.