  Motivated by a partition inequality of Bessenrodt and Ono, we obtain
analogous inequalities for $k$-colored partition functions $p_{-k}(n)$ for all
$k\geq2$. This enables us to extend the $k$-colored partition function
multiplicatively to a function on $k$-colored partitions, and characterize when
it has a unique maximum. We conclude with one conjectural inequality that
strengthens our results.
