  The classical rearrangement inequality provides bounds for the sum of
products of two sequences under permutations of terms and show that similarly
ordered sequences provide the largest value whereas opposite ordered sequences
provide the smallest value. This has been generalized to multiple sequences to
show that similarly ordered sequences provide the largest value. However, the
permutations of the sequences that result in the smallest value are generally
not known. We show a variant of the rearrangement inequality for which a lower
bound can be obtained and conditions for which this bound is achieved for a
sequence of permutations. We also study a generalization of the rearrangement
inequality and a variation where the permutations of terms can be across the
various sequences. For this variation, we can also find the minimizing and
maximizing sequences under certain conditions. Finally, we also look at
rearrangement inequalities of other objects that can be ordered such as
functions and matrices.
