Applications of Hankel's determinant for analytical functions

In this thesis, some versions of Schwarz lemma are given for different classes which were defined on the unit disk. New inequalities for analytic functions were obtained using Hankel's determinant and obtained an upper bound for this determinant. Moreover, estimations from below of the modulus of the angular derivative were given assuming the angular limit at the boundary point of the unit disk. Finally, the sharpness of the inequalities obtained in the presented theorems were proved.