Schwarz-Christoffel Polygon Formula,the Upper Half Plane Capacity And The Growth Estimate Of The Coefficient

The Loewner differential equation is an important content of univalen-t functions, and it is proved to be one of the most useful tools in solving the extremal problems of univalent functions. With a view to investigating the Scaling limit of some models in statistical physics, Schramm established a stochastic version of differential equation in 1999 referred to as Stochastic Loewner Evolution, denoted(SLE). With the rapid development since its es-tablishment this theory has become an important research topic in the fields of statistical physics, probability theory and conformal invariant theory. Based on complex analysis and stochastic analysis, a large number of international leading mathematicians and physicists join the research of the frontier prob-lems in statistical physics using a research tool of the Loewner Evolution.H-hull is a suitable compact set K falling in the upper half plane, which is an important research object of Loewner differential equation. Sometimes H-hull is also a mathematical description of physical model, corresponding to the fact that the upper half plane capacity heap (K) is an important content in SLE theory. One of the purposes of this thesis is to calculate its upper half plane capacity heap (K) accurately when H-hull K is circular arc tangent to the real axis.In this thesis, the other research problem is the growth estimate of the coefficient, which is also a hot issue that (function theory) analysis scientists focus on. Avkhadiev, Pommerenke Wirths introduced the definition of concave functions in [20], [21]:If f is a conformal mapping which maps a unit circle (analysis) on the outside of a convex set, we call f a concave function. Study in this thesis concludes the coefficient estimate of a special kind of concave function.This thesis consists of four chapters, the specific arrangement is as follows.In the first chapter, it mainly introduces the H-hull, some research back-ground and significance of the upper half plane capacity, and the current re-search status of coefficient estimates. At the same time, it introduces the main conclusions of this thesis.In the second chapter, based on the Schwarz-Christoffel polygonal for-mula, this thesis gives the accurate value of the upper half plane capacity heap (K) of circular arc as H-hull. The arc starts from the real axis and is tangent to the real axis, then all fall in the upper half plane except the origin.In the third chapter, this thesis calculates the accurate value of the upper half plane capacity heap (K) of two circular arcs as H-hull. The two arcs starting from real axis and tangent to the real axis, then all fall in the upper half plane except the origin.In the fourth chapter, the paper studies a special kind of concave function: 0<\z\< 1. It is an analytic conformal mapping in D\{0}, and C\f(D)= G is a convex polygon (simply connected). This thesis concludes the coefficients growth estimates of am in f.