Certain Subclasses of Bi-Univalent Functions Associated with Generalized Mersenne Polynomials

The study of bi-univalent functions, whose defining property requires both the function and its inverse to be univalent in the unit disk, has received considerable attention due to its intrinsic analytical complexity. In this paper, a novel subclass of analytic and bi-univalent functions is formulated in relation to generalized Mersenne polynomials. These polynomials are employed as an effective tool for constructing operators related to this class of functions. The main focus of the paper is to obtain coefficient estimates for the Taylor–Maclaurin expansion, with particular emphasis on the initial coefficients. Using subordination techniques and standard methods from geometric function theory, several Fekete–Szegö type inequalities are derived. In addition, a number of special cases and related results are discussed to establish connections with previously known subclasses. The results presented here extend and unify various earlier coefficient inequalities in the literature and highlight the applicability of Mersenne polynomials in coefficient problems for bi-univalent function classes.