Abstract :
We show that if w y and x z are four vectors in Rn, then a number of Schur majorizations hold between “symmetrized” vector functions of w,x,y and z, e.g., (wi+xj)i,j (yi+zj)i,j where the left-hand expression means the vector of dimension n2 consisting of all sums wi+xj of the co-ordinates of w and x, arranged in lexicographic order. Among other things, we get vector and matrix versions of Muirhead’s theorem for scalar inequalities. From the vector inequalities follow many scalar inequalities for “symmetrized” sums, some of which are scattered through the inequality literature.
In Section 2, applications are given to matrix inequalities for tensor products, e.g., if A,B and C are Hermitian and λ(A) λ(B), then λ(A C) λ(B C).
