Modulus Growth Bounds On Analytic Functions

The Schwarz Lemma is one of the most central results in complex analysis. Since Ahlfors applied the curvature to making researches on Schwarz Lem-ma, some other geometric quantities were used to do it and many great results were yielded, such as Landau and Toeplitz applied the notions of radius, diameter, area to analytic functions of the unit disk and some results about monotonicity and the growth bound of modulus have been obtained.In this dissertation, some bounds of the distance of f from its Taylor approxima-tion are obtained, where the analytic functions f in D satisfy Rad f(D) ≤1 or Diam f(D) ≤2.