Quadrangular Ratio Metrics And Quasiconformal Mappings

Metrics play active and important roles in the related studies in mathe-matics.The investigation on metrics not only has its own research value,but also,as research tools,is useful in the study of other research fields.Hence,the study on metrics gains more and more attention.The main research ob-ject of this thesis is the quadrangular ratio metrics raised by Aseev,which is a special kind of angular metrics.Quadrangular ratio metrics are a kind of hyperbolic type metrics determined by two boundary points.Up to now,there are not many studies on these metrics in the literatures.Hence,there are a lot of problems related to these metrics,which need to be investigated further.The purpose of this thesis is to discuss the properties of quadrangular ratio metrics,and then,as applications,we study the properties of quasiconformal mappings.This thesis consists of four chapters,and its arrangement is as follows.In the first chapter,we will introduce the background of the research prob-lems and give the statements of the main results.In the second chapter,we will study the comparison theorems of quadran-gular ratio metrics in Rn.Firstly,we establish the comparison theorem between quadrangular ratio metrics and Seittenranta metrics,which is the quadrangular ratio metrics version of the corresponding result obtained by Vuorinen et al in 2015 and 2017.Then,as an application,we establish comparison theorems between quadrangular ratio metrics and other hyperbolic type metrics.In the third chapter,we study the relation between quadrangular ratio met-rics and quasiconformal mappings,quasimobius mappings,and we obtain the properties of quadrangular ratio metrics under quasiconformal mappings and quasimobius mappings.These results are generalizations of the corresponding results obtained by Seittenranta in 1999.Then we get the property of quadran-gular ratio metrics under solid mappings.In the fourth chapter,we show the relations between quadrangular ratio metric balls and Euclid metric balls,which is a generalization of the corre-sponding result obtained by Seittenranta in 1999.As an application,we obtain a relationship between Lipschitz mappings in the quadrangular ratio metrics and quasiconformal mappings.