On Some Inequalities Related To Operator Mean And Monotone Functions

In this paper, based on the operator power mean and Karcher mean and their prop-erties, we shall introduce some operator inequalities involving positive linear map and talk about the relationship between operator inequalities and monotone functions. We separate this paper into three parts in order to introduce the related problem.The first part is introduction. We mainly introduce the notions and properties of power mean and Karcher 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, we use some properties of the power mean and Karcher mean to obtain some operator inequalities combining positive linear map and we show these inequalities are order preserving under p-th powering. Besides, we give extensions about some important operator inequalities in the same way.The third part introduce the relations between the operator inequalities and the op-erator monotone functions. By using the fundamental conclusions based on majorization. namely, product lemma and product theorem for operator monotone functions, we can give some operator inequalities. This result contains the Furuta inequality, which has a huge impact on positive operator theory.