The permutation theorem is very useful when dealing with inequalities between sums of products of the two real number sequences. In numerous cases inequalities otherwise difficult to prove can be proven almost automatically.

1. Definition (Similarly and oppositely ordered sequences)
Two sequences of real numbers (a1, . . . , an) and (b1, . . . , bn) are similarly ordered if and only if for each
pair (i, j), where 1 ≤ i, j ≤ n, we have
(ai − aj)(bi − bj) 0 (1)
in other words
(ai aj ∧ bi bj) ∨ (ai aj ∧ bi bj)
Analogically, the aforementioned sequences are oppositely ordered if and only if the inequality (1) is reversed.
For the purpose of the theorem formulated below, let’s introduce the following notation
n
k=1
akbk = a1b1 + a2b2 + · · · + anbn =
a1 a2 · · · an
b1 b2 · · · bn
Theorem (Inequalities between sums of the products of the sequences elements) :
If the sequences of real numbers (a1, . . . , an) and (b1, . . . , bn) are similarly ordered then for
any permutation (b1, . . . , bn) of the given sequence (b1, . . . , bn) we have
n
k=1
akbk =
a1 a2 · · · an
b1 b2 · · · bn
a1 a2 · · · an
b1 b2 · · · bn
=
n
k=1
akbk (2)
If the sequences of real numbers (a1, . . . , an) and (b1, . . . , bn) are oppositely ordered then for
any permutation (b1, . . . , bn) of the given sequence (b1, . . . , bn) we get the reversed version of
the inequality (2).
c 2015/10/13 22:41:36, Mikołaj Hajduk 1 / 1