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. ‎Morever‎, ‎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.