Coefficient Problems for Bi-Univalent Functions via $q$–Rabotnov Kernels and $q$–Fibonacci Subordination

Abstract

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 $|\alpha_{2}|$ and $|\alpha_{3}|$, 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.