International Journal of Neutrosophic Science IJNS 2690-6805 2692-6148 10.54216/IJNS https://www.americaspg.com/journals/show/3796 2020 2020 Department of Mathematics, Zarqa University, 2000 Zarqa, 13110, Jordan Waleed Waleed This paper introduces two new classes of bi-Bazilevic and bi-univalent functions that are defined using Borel distribution and Ruscheweyh operator, which also associated with Legendre polynomials and modified Sig-moid function within the open unit disk D. This paper explores the characteristics and behaviors of these functions, we find estimates for the modulus of the initial Taylor series coefficients a2 and a3 for functions within our newly defined classes and some of their various subclasses. Moreover, this paper explores the classical Fekete-SzegÖ functional problem concerning functions f that are classified within our specific classes. Additionally, we obtain the classical Fekete-SzegÖ inequalities of functions belonging to these classes and some of their various subclasses. 2025 2025 242 258 10.54216/IJNS.260317 https://www.americaspg.com/articleinfo/21/show/3796