(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Motivated by the interplay between q–calculus and geometric function theory, this paper introduces and investigates a new subclass of bi-univalent functions associated with shell-like domains generated through the q–Rabotnov function and the q–analogue of Fibonacci numbers. A central contribution of this work is the definition of a novel q–derivative operator, constructed via convolution with kernels involving the q–Rabotnov function. Employing the subordination principle, we derive sharp coefficient estimates for the initial Taylor–Maclaurin coefficients |α_2| and |α_3|, and establish Fekete–Szegö-type inequalities for the proposed class. The results obtained here unify and extend several recent contributions in the theory of bi-univalent functions, while also highlighting the role of q–special functions in generating new analytic structures. These findings enrich the structural understanding of bi-univalent functions and suggest future directions involving operator theory, convolution structures, and further applications of q–calculus in complex analysis.