Study On Equivalence Of Order Preserving Inequalities For Hiai Type Log-majorization Theory

Operator log-majorization theory plays an important role in some branches of mathematic-s,For some types of inequalities,operator log-majorization can be a very useful and powerful theory.In addition,it is often helpful to derive an inequality by using operator log-majorization methods to understand inequality more deeply.In this paper,we study the mutual transforma-tion between the negative power Furuta inequality and the logarithmic superior superinequality.At the same time,we give different methods to prove the theorems,make the content of the theorem more rich and concrete.The contents are as follows:First of all,some studies are made on the theory of logarithmic superiority of Hiai type.We will show the extension of the Hiai type log-superior superinequality,which is essentially that the Hiai log-superior superinequality is equivalent to two forms of the Furuta inequality with negative powers.Secondly,we discuss the extension of pseudo-power mean and Hiai type log-superior su-perinequality,and then we discuss the Hiai type log-superior superinequality with A(?)B as the lower bound.At the same time,the derivation between it and Furuta inequality with negative power type is proved.Finally,The main work of this paper is summarized,and the future work is prospected.