Majorizations of Hadamard products of matrix powers

This paper presents a number of intervening terms for the weak majorization (n)(i = k) Pi lambda(i)(A circle B) greater than or equal to (n)(i = k) Pi lambda(i)(AB) for k = 1, 2,...,n, where A, B > 0. More specifically, it proves for r is an element of (0, 1] that (n)(i = k) lambda(i)(A circle B) greater than or equal to (n)(i = k) Pi lambda(i)(1/r)(A(r) circle B-r) greater than or equal to (n)(i = k) Pi e(lambda i(I circle (log A + log B)) greater than or equal to (n)(i = k) Pi e(lambda i(log A + log B) greater than or equal to (n)(i = k) Pi lambda(i)(1/r)(A(r)B(r)) greater than or equal to (n)(i = k) Pi lambda(i)(AB) for k = 1,...,n. It also shows that behind many of these inequalities are stronger ones. (C) 1998 Elsevier Science Inc.