It is shown that if
a
(
t
)
=
(
a
1
(
t
)
,
a
2
(
t
)
,
…
,
a
n
(
t
)
)
,
t
=
1
,
…
,
m
{a^{(t)}} = (a_1^{(t)},a_2^{(t)}, \ldots ,a_n^{(t)}),t = 1, \ldots ,m
, are nonnegative
n
n
-tuples, then the maxima of
∑
i
=
1
n
a
i
(
1
)
a
i
(
2
)
⋯
a
i
(
m
)
\sum \nolimits _{i = 1}^n {a_i^{(1)}a_i^{(2)} \cdots a_i^{(m)}}
of
∏
i
=
1
n
min
t
(
a
i
(
t
)
)
\prod \nolimits _{i = 1}^n {{{\min }_t}(a_i^{(t)})}
and of
Σ
i
=
1
n
\Sigma _{i = 1}^n
min
(
a
i
(
t
)
)
(a_i^{(t)})
, and the minima of
∏
i
=
1
n
(
a
i
(
1
)
+
a
i
(
2
)
+
⋯
+
a
i
(
m
)
)
\prod \nolimits _{i = 1}^n {(a_i^{(1)} + a_i^{(2)} + } \cdots + a_i^{(m)})
, of
∏
i
=
1
n
max
t
(
a
i
(
t
)
)
\prod \nolimits _{i = 1}^n {{{\max }_t}(a_i^{(t)})}
and of
∑
i
=
1
n
max
t
(
a
i
(
t
)
)
\sum \nolimits _{i = 1}^n {{{\max }_t}(a_i^{(t)})}
are attained when the
n
n
-tuples
a
(
1
)
,
a
(
2
)
,
…
,
a
(
m
)
{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
)
(0,1)
-matrices is given.