aspg-21-3796-1791472342 20261008151222 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group International Journal of Neutrosophic Science IJNS 2692-6148 2690-6805 2025 26 3 Waleed Al-Rawashdeh Department of Mathematics, Zarqa University, 2000 Zarqa, 13110, Jordan 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 242 258 3796 https://creativecommons.org/licenses/by/4.0/ 10.54216/IJNS.260317 https://www.americaspg.com/journal/21/article/3796 https://www.americaspg.com/storage/41744982712.pdf