Coefficient Estimates for New Subclasses of Bi-Univalent Functions Involving Generalized Bivariate Fibonacci-like Polynomials

<ns3:p>Background The study of bi-univalent functions has gained considerable attention due to its relevance in geometric function theory, particularly in determining bounds for the initial coefficients of analytic functions. Previous research has introduced several subclasses of bi-univalent functions using tools such as convolution, subordination, differential subordination, and various families of special polynomials. Methods Motivated by these developments, this work constructs new subclasses of bi-univalent functions by employing generalized Fibonacci polynomials. The analytic properties of these polynomials are used to define the classes and derive the corresponding functional relations needed for coefficient estimation. Results For the newly introduced subclasses, bounds for the initial coefficients are established. Furthermore, Fekete–Szegö inequalities are obtained for each subclass, extending previously known results and demonstrating the influence of the generalized Fibonacci structure on coefficient behavior. Conclusions The results contribute to the ongoing development of bi-univalent function theory by providing broader subclasses and sharper estimates. The use of generalized Fibonacci polynomials offers a flexible analytical framework that can be further applied to related classes in future studies.</ns3:p>