Certain Subclasses of Univalent and Bi-Univalent Functions Involving New Operator

The authors introduce novel subclasses within the univalent and bi-univalent function classes in the open unit disk . Using a newly defined operator, the study explores the properties of holomorphic bi-univalent functions in , focusing on parameters such as the Taylor-Maclaurin coefficients , the Fekete-Szegö functional, and the second-order Hankel determinant. These parameters are critical for understanding the geometric and holomorphic characteristics of these functions. The investigation emphasizes the significance of the proposed operator and subclasses under varying parameter conditions, contributing valuable insights to geometric function theory. Derived corollaries further refine existing results, demonstrating practical applications and expanding the theoretical framework of the field.