Some Results On Weighted Mean Inequalities Combinatorial Mean Inequalities And Singular Values Of Matrices

In this thesis,we study three topics about weighted mean inequalities new combinatorial mean inequality and singular values of matrices.Our main results are divided into three parts.First,we prove following a new mean inequality,i.e.,(?)where ai,bi are positive real numbers(i=1,2,...,n)and p,q are arbitrary real numbers by using the elementary inequality(?) here fi:E(?)R?R,(i=1,2,...,n)and constructing auxiliary functions.Using this inequality,we deduce the harmonic-geometric-arithmetic square mean inequality.As an application,a new characterization of unitary matrix is given by using the formula and numerical radius.Second,a new combinatorial arithmetic-geometric mean inequality is established by using multivariate polynomial theorem and we solve an No.12066 open problem about determinant of positive definite matrix in the American Mathematical Monthly.At last,we obtain a characterization of pair matrices such that sj(A)?sj(B),j=1,…,n where s,(T)denotes the j-th largest singular values of T.Using this characterization,we give a new proof of Zhan's result related to singular value inequality of differences of positive semidefinite matrices.