Bounds on Hankel Determinants with Fekete-Szegö Parameter for Bazilević Functions

<ns7:p>Background In Geometric Function Theory, a central area of complex analysis, researchers study the geometric properties of analytic and univalent functions in the unit disk. A significant part of this work involves defining some subclasses of functions and investigating their properties, such as coefficient estimates and Hankel determinants, which reveal important geometric information. This paper introduces a comprehensive subclass of analytic-univalent functions that generalizes well-known families like Yamaguchi and starlike functions within the broader Bazilević framework. Methods and Results Using the theory of Ma-Minda functions and the principle of subordination, sharp bounds for the initial coefficients, the Fekete-Szegö functionals with parameters, and the Hankel determinants are established for this subclass. The results are proven to be sharp, meaning they are the best possible. Furthermore, it is shown that this general class reduces to several previously known function families for specific parameter values, demonstrating its wide applicability. Conclusions This research successfully defines and analyzes a comprehensive subclass of analytic and univalent functions. The obtained sharp bounds for coefficient-related problems generalize and extend existing results in the literature. The work provides a unified framework for studying various function families, contributing valuable insights and tools for further exploration in Geometric Function Theory.</ns7:p>