Bi-Univalent Function Families Involving q-Rabotnov Function and q-Analogues of Fibonacci Numbers

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 curves defined via the $\q$-Rabotnov function and the $\q$-analogue of Fibonacci numbers. By employing the subordination principle, we derive coefficient bounds for the initial Taylor--Maclaurin coefficients, specifically $|a_{2}|$ and $|a_{3}|$, and further establish sharp Fekete--Szegö type inequalities for the proposed function class. Our results not only extend and generalize several recent contributions in the theory of bi-univalent functions but also highlight novel connections between $\q$-special functions, shell-like domains, and analytic inequalities. The findings presented herein contribute to a deeper understanding of the structural properties of bi-univalent functions and open avenues for future applications in operator theory and related analytic frameworks.