Abstarct: In this dissertation‎, ‎we find coefficient estimates applying‎ the Faber polynomial expansions and using a new method of the previous results for some subclasses of‎ analytic bi-univalent functions‎, ‎which are defined by subordinations‎ in the open unit disk‎. ‎The coefficient bounds presented in this‎ paper would generalize and improve some recent works appeared in the‎ literature‎. ‎Furthermore‎, ‎we obtain the second Hankel determinant with‎ a different method for some certain classes of analytic bi-univalent function‎ which are defined by subordinations in the open unit disk that the‎ presented results in this field improve or generalize the recent works of‎ ‎other authors‎.‎