Schwarz's lemma and its applications on unit disk

In this thesis, an improved version of the classical Schwarz lemma is presented, extending its applications to a certain class of analytic functions defined on the unit disk. The study includes new limit versions of the Schwarz lemma, which establish exact inequalities for the angular derivatives of these functions at the boundary point under certain conditions. These inequalities are obtained for the first few coefficients of the Taylor expansion of and their exactness is demonstrated by giving examples.