Some Properties Of The Scale Invariant Cassinian Metric And A Gromov Hyperbolic Metric

The thesis is devoted to studying some properties for the scale invariant Cassinian metric and a Gromov hyperbolic 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 scale invariant Cassinian metric and the Gromov hyperbolic metric.In chapter 2,we study the geometric properties of the scale invariant Cassinian metric,and give the formulas in some special cases and obtain some upper and lower bounds for the scale invariant Cassinian metric in the unit disk or in the upper half plane.We obtain sharp inequalities between the scale invariant Cassinian metric and the hyperbolic metric as well as some other hyperbolic-type metrics.We prove sharp distortion inequalities of the scale invariant Cassinian metric under M(?)bius transformations.In chapter 3,we obtain sharp comparison inequalities between the Gromov hyperbolic metric and the hyperbolic metric,the distance ratio metric as well as some other hyperbolic-type metrics.We also show sharp distortion inequalities of the Gromov hyperbolic metric under M(?)bius transformations.In chapter 4,we summarize the main work of this thesis and put forward some problems for further study.