On The Study Of Several Generalizations Of Several Matrix Function Inequalities And Its Applications

We in this paper establish some new inequalities via different matrix functions,which extend some recent results.In Chapter 2,considering the numerical ranges of A contained in S? ={rei?|r ? 0,|?| ??,? ?[0,?/2)}and S'?={rei?|r?0,0????,??[0,?/2)},respectively,we prove some better upper bounds of Rotfel'd type theorems.In addition,we give a new equivalent form of an extension of the trace inequality of Rotfel'd theorem due to Lee.Furthermore,some recent Rotfel'd type theorem extensions are all equivalent via the above equivalent proposition.In Chapter 3,we prove inequalities on non-integer powers of generalized matrices functions on the sum of positive semi-definite matrices.For example,for any generalized matrix function d,positive semi-definite matrices A.B,C,and positive integer r,it is known that d(A + B + C)r + d(A)r + d(B)r + d(C)r ? d{A + B)r + d(B + C)r + d(A + C)r.We show that the same inequality holds for any real number r? {1} ?[2,?).A gen-eral scheme is introduced to prove more general inequalities involving m positive semi-definite matrices for m ? 3 that extend the results of other authors.In Chapter 4,we obtain several new inequalities for positive linear maps,which also generalize some recent results.As applications,we· give simple proofs of some known results;· present a better lower bound of spectral condition number for any nonsingular ma-trix with its entries.Numerical experiments show the efficiency of our method.