Boundary Schwarz Lemma for a certain class of analytic functions

In general, in this thesis, the boundary behavior of analytic, bounded functions in the unit disc is examined, and different versions of the Boundary Schwarz Lemma are discussed. This thesis includes three main sections. The first section is the introduction and in this section, a general knowledge of the literature is given. Also, basic definitions, theorems, and corollaries are presented to make the obtained results more clear. In the second section, new results are mentioned for a certain class of analytic functions in the unit disc. In this section, a version of Schwarz Lemma is obtained for the functions belonging to that class. Next, on the boundary point of the unit disc, the existence of the angular limit of the function belonging to the class discussed is considered and the inequalities which are estimated from below the modulus of the angular derivative are obtained on the boundary. In the theorems in this section, different cases about the zeroes of the function are examined separately and the sharpness of some of the obtained inequalities is proved. In the third section, which is the last section, the examples of the functions belonging to the examined class are given.