Some inequalities for singular values of matrix products

Let α1(C) ≥ … ≥ αn(C) denote the singular values of a matrix C ε n×m, and let 1 ≤ i1 < … < ik ≤ n, 0 < r ε . The main results are Σt=1kΣitr(A)Σn−it+1(B), where A ε p×n, B ε n×m. We also consider the cases for the product of three matrices and more.