Discussion And Generalization Of Problems Via The Furuta Inequality

Operator inequality is an important branch of operator theory.In 1934, L(o|¨)wner established the L(o|¨)wner-Heinz inequality,which is a foundation of operator inequality theory.After that,Furuta inequality,a famous operator inequality,is proposed by Japanese mathematician Furuta in 1987 and based on the L(o|¨)wner-Heinz inequality.Now,Furuta inequality has played an important role in operator inequality,operator equation and the study of mathematics,physics.This dissertation mainly studies the remainder problem of Furuta inequality by software Mathematica 5.0;With an elementary method,two groups operator functions are obtained,which enrich the content of operator monotone functions.Then the Furuta inequality and Grand Furuta inequality are generalized by the theory of operator mean and the theory of operator monotonicity.In the first chapter,we introduce the Furuta inequality and its remainder problem,according to concrete examples we explore the probability of remainder problem by software Mathematica 5.0,then we get the result that the Furuta inequality is positive,this conjecture will help us recognize the remainder problem further;In addition,the Young inequality is proved by partitioned matrix. The second chapter mainly concentrates on obtaining some operator monotone function.Firstly,we introduce its definition,then construct two groups of operator functions by an elementary method,and extend the function f(t)= t(t+2)log(t+2)/(t+1)~2 to linear situation,at last we discuss some results about operator monotone function under the complex order and conjugate complex order.The third chapter discusses some functions in the form ofα-power mean.We first demonstrate the geometric structure of Furuta inequality and generalize the ordinary order to the order defined by A~4≥(A~2B~2A~2)~2/3,and discuss its properties; Furthermore,we extend these properties to the order A~2m≥(A~mB~2A~m)~m/m+1 and obtained some new results.In the fourth chapter,we mainly give some results about Grand Furuta inequality.In the last chapter,we give a summary of this paper and list some fields worthy of studying about the Furuta inequality and expect more results will be get through our joint efforts.