Applications of Gegenbauer Polynomials for Subfamilies of Bi-Univalent Functions Involving a Borel Distribution-Type Mittag-Leffler Function

In this research, a novel linear operator involving the Borel distribution and Mittag-Leffler functions is introduced using Hadamard products or convolutions. This utilized to develop new subfamilies of bi-univalent via principle subordination with Gegenbauer orthogonal polynomials. The investigation also focuses on estimation coefficients |a?|(?=2,3) Fekete–Szegö inequality for belonging these functions. Several corollaries implications findings are discussed. Overall, study presents approach constructing provides valuable insights further research in area.