Uniformly John Domains And Quasihyperbolic Cosine Inequalities

In order to investigate the properties of injectivity and approximation of bi-Lipschitz mappings, in 1961, John introduced a class of domains in C which was named as John domians by Sarvas and Martio in 1978. In order to gen-eralize Ahlfors' study on the properties of injectivity of conformal mappings in quasidisks, in 1978, Martio and Sarvas introduced a class of new domains which are uniform domains. As the generalization of uniform domains, Vaisala defined the uniformly John domains. In 1976, as the generalization of the hyperbolic metric, Gehring and Palka introduced the quasihyperbolic metric. These do-mains and the quasihyperbolic metric are closely relative to the research in the fields of analysis and geometry, all of which creat much attention.The study of this thesis includes two parts. In the first part, we charac-terize uniformly John domains in term of quasihyperbolic metric, jD* metric,ρ-Apollon metric. In the second part, we discuss the corresponding properties of quasihyperbolic geometry and the hyperbolic cosine inequality. This thesis consists of three chapters. It is arranged as follows.In chapter one, we introduce the background of our research problems and state our main results.In chapter two, we study the relations betweenρ-Apollon metric and uni-formly John domains. We get a necessary and sufficient condition for uniformly John domains by using p-Apollon metric. Furthermore, we construct two ex-amples to show that the corresponding constant in our result can't be removed.In chapter three, the attention is concentrated on some properties of quasi-hyperbolic geometry. We get corresponding Pappus' formula, Stewart theorem, Lagrange theorem and establish a hyperbolic cosine inequality which gives an affirmative answer to one of open problems raised by Klen.