Some Refined Operator Mean Inequalities And YOUNG Type Inequalities

In this paper, based on the several operator means and their properties, we shall introduce some refined operator inequalities involving positive linear map and study the two arbitrary means between arithmetic and geometric means. Finally, we give some refined Young type inequalities with matrix mean. We separate this paper into three parts in order to introduce the related problem.The first part is introduction. We mainly introduce the definitions and properties of the several operator means 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 perator means in the first part to obtain some refined operator inequalities.The third part we refine some operator inequalities and matrices inequalities for the Hilbert-Schmidt norm. First, the refined Young inequalities for scalars are given. Then,based on these inequalities, operator inequalities, matrices inequalities on the Hilbert-Schmidt norm and determinant inequalities are established.