On Some Inequalities Related To Operator Mean And Matrix Mean

In this paper, based on the operator geometric mean, Heinz mean and their properties,we shall introduce some operator inequalities involving positive linear map and talk about some matrix inequalities. We separate this paper into two parts in order to introduce the related problem.The first part is introduction. We mainly introduce the notions and properties of geometric mean and Heinz mean and research backgrounds of this paper. Furthermore, we also introduce several theorems and knowledge that are referred frequently in this paper.The second part is an important part in this paper, based on the operator Diaz-Metcalf type inequality, we use some properties of the geometric mean and Heinz mean to obtain some operator inequalities combining positive linear map. And based on the order preserv-ing under p-th powering of ?-geometric mean inequality, we give extensions about some operator inequalities related to Heinz mean. Besides, by the properties of k-th symmetric tensor power and the relationship between k-th symmetric tensor power and singular value,we also get some matrix mean inequalities.