Consider all ordinary arithmetic
and geometric means of n real nonnegative numbers taken k at a time. Let αk be the
geometric mean of all the arithmetic means and γk the arithmetic mean of all the
geometric means. It is proved that αk increases with k,γk decreases, and γh≦ αk if
h + k > n. These results are generalized to mixed means of any real orders.
Comparison of αk and γk with elementary symmetric functions suggests a
conjecture.