Comprehensive Subfamilies of Bi-Univalent Functions Involving a Certain Operator Subordinate to Generalized Bivariate Fibonacci Polynomials

This paper introduces novel subfamilies of analytic and bi-univalent functions in Ω=ς∈C:|ς|<1, defined by applying a linear operator associated with the Mittag–Leffler function and requiring subordination to domains related to generalized bivariate Fibonacci polynomials. The proposed framework provides a unified treatment that generalizes numerous earlier studies by incorporating parameters controlling both the operator’s fractional calculus features and the domain’s combinatorial geometry. For these subfamilies, we establish initial coefficient bounds (d2, d3) and solve the Fekete–Szegö problem (d3−ξd22). The derived inequalities are interesting, and their proofs leverage the intricate interplay between the series expansions of the Mittag–Leffler function and the generating function of the Fibonacci polynomials. By specializing the parameters governing the operator and the polynomial domain, we show how our main theorems systematically recover and extend a wide range of known results from the literature, thereby demonstrating the generality and unifying power of our approach.