Rearrangements

Abstract:

It is shown that if ${a^{(t)}} = (a_1^{(t)},a_2^{(t)}, \ldots ,a_n^{(t)}),t = 1, \ldots ,m$, are nonnegative $n$-tuples, then the maxima of $\sum \nolimits _{i = 1}^n {a_i^{(1)}a_i^{(2)} \cdots a_i^{(m)}}$ of $\prod \nolimits _{i = 1}^n {{{\min }_t}(a_i^{(t)})}$ and of $\Sigma _{i = 1}^n$ min $(a_i^{(t)})$, and the minima of $\prod \nolimits _{i = 1}^n {(a_i^{(1)} + a_i^{(2)} + } \cdots + a_i^{(m)})$, of $\prod \nolimits _{i = 1}^n {{{\max }_t}(a_i^{(t)})}$ and of $\sum \nolimits _{i = 1}^n {{{\max }_t}(a_i^{(t)})}$ are attained when the $n$-tuples ${a^{(1)}},{a^{(2)}}, \ldots ,{a^{(m)}}$ are similarly ordered. Necessary and sufficient conditions for equality are obtained in each case. An application to bounds for permanents of $(0,1)$-matrices is given.