Abstarct: In this thesis, we introduce the subclasses of bi-univalent functions by using the real positive part functions and obtain the estimates for the initial coefficients a_2, a_3 and a_4 for functions of mentioned subclasses. The coefficient estimates would generalize and improve some recent works. Moreover, we define the subclass of the m-fold symmetric bi-univalent functions, and we find the estimates of the initial coefficients a_{m+1} and a_{2m+1} by using real positive part functions. The coefficient estimates would generalize and improve some recent works. Also, The Fekete–Szegő functional is derived for functions that belong to these subclasses.