Representation of Bi-Univalent Functions to Lucas Balancing Polynomials with Geometric Properties and Coefficient Bounds

In this paper, we introduce and analyze a new subclass of bi-univalent functions defined in the open unit disk, associated with Lucas and Lucas balancing polynomials. By employing the Taylor-Maclaurin series expansion, precise bounds for the second and third coefficients, |a₂| and |a₃| , are obtained. These estimates lead to important geometric interpretations related to the distortion, growth, and structural behavior of the functions near the origin. Furthermore, the mapping characteristics of these functions are examined in connection with existing bi-univalent subclasses. Special emphasis is placed on deriving a Fekete-Szegö type inequality for the proposed class, thereby extending earlier contributions in the field. The findings presented here are valuable for applications in geometric function theory and areas like fluid mechanics, conformal mappings, and engineering models where the analytic structure of bi-univalent functions plays a significant role.