Comprehensive Subfamilies of Bi-Univalent Functions Defined by Error Function Subordinate to Gegenbauer Polynomials

This work explores coefficient estimates for analytic functions in a symmetric domain. Building on recent studies that use polynomials such as Lucas and Legendre polynomials to bound the Maclaurin coefficients |a2| and |a3|, we turn to Gegenbauer polynomials. By applying the imaginary error function and subordination techniques, we derive sharp bounds for |a2| and |a3| and the Fekete-Szegö functional for two extensive new subclasses. Our general theorems also yield several novel special cases, demonstrating the breadth of our results.