The Schwarz Lemma At The Boundary Of The Strip

The Schwarz lemma is one of the most important contents in theory of functions of a complex variable, which can be widely applied to other subjects. For the analytic functions of the unit disk into the unit disk, the properties of these functions in the unit disk and at boundary of the unit disk have been studied for many years,meanwhile, the Schwarz lemma and the Schwarz lemma at the boundary have been obtained, respectively.In this dissertation, the analytic functions of the unit disk into the strip 0 <Imf(z) < π are considered, and the corresponding Schwarz lemma and the Schwarz lemma at the boundary are obtained. In chapter two, we introduce some properties of the analytic maps and the Poincare metric, and then present the development and application of the Schwarz lemma and the Schwarz lemma at the boundary. Moreover,main results are provided in chapter three.