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If a1 ≥ a2 ≥ ... ≥ an and b1 ≥ b2 ≥ ... ≥ bn then
| n | n � k=1 |
akbk ≥ | ( | n � k=1 |
ak | )( | n � k=1 |
bk | ) |
The proof is that the RHS can be written as the sum of n different
sums,
(a1b1 + a2b2 + ... + an-1bn-1
+ anbn) +
(a1b2 + a2b3 + ... + an-1bn
+ anb1) +
...
(a1bn + a2b1 + ... + an-1bn-2
+ anbn-1)
The first of these n sums is at least as big as each of the others by the
rearrangement inequality, and n
times this first sum is the LHS, so the LHS is at least as big as the RHS.
The AM-GM Inequality: the Arithmetic Mean of positive numbers is always greater than the Geometric Mean. This is proved using Jensen's Inequality.
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