Pointwise domination of matrices and comparison of ℐ_p norms

Let p be a real number in [1, ∞) which is not an even integer. Let N = 2[p/2] + 5. We give examples of N X N matrices A and B, so that |aij| ≤ bij but Tr([A*A]p/2) > Tr([B*B]p/2).