Geometric Properties Of The Triangular Ratio Metric And H＿{D,c}-Metric

The thesis is devoted to studying some properties for the triangular ratio metric and the h_D,c-metric.In chapter 1,we introduce the background and significance of this work and definitions about several related metrics,and present research progress of the triangular ratio metric and the h_D,c-metric.In chapter 2,we study the geometric properties of the triangular ratio metric,and give the formulas in some special points and obtain upper and lower bounds for the triangular ratio metric in the unit disk.We obtain sharp inequalities between the triangular ratio metric and the hyperbolic metric as well as some common hyperbolic-type metrics.We also show distortion inequalities of the triangular ratio metric under M（?）bius transformations.We study inclusion properties of the triangular ratio metric balls in proper subdomains of Rn with other related metric balls.In chapter 3,we show distortion inequalities of the h_D,c-metric under M（?）bius transformations from a punctured ball onto another punctured ball.In chapter 4,we summarize the main work of this thesis and put forward some problems for further study.