Inequalities - Holder's inequality

§ Suppose that (a), (b), …, (l) are m sets each of n numbers.
Then (a)+(b)+...+(l)<(a+b+...+l), unless either (1) every two of (a), (b), …, (l) are proportional, or (2) there is a ν such that aν=bν=...=lν=0.

§ If α,β,...,λ are positive and α+β+...+λ=1, then

∑aαbβ...lλ<(∑a)α(∑b)β...(∑l)λ,

unless either (1) the sets (a),(b),...,(l) are all proportional, or (2) one set is nul.

§ If r,α,β,...,λ are positive and α+β+...+λ=1, then

¯r(ab...l)<¯r/α(a)¯r/β(b)...¯r/λ(l)

unless (a1/α),(b1/β),...,(l1/λ) are proportional or one of the factor on the right-hand side is zero. If r<0, the inequality is reversed.

§ Any two real numbers k and k′ are conjugate if they are satisfied any one of the three following equalities:

1. k′=kk−1, with k≠1
2. (k−1)(k′−1)=1
3. 1k+1k′=1 with k≠0, and k′≠0

Suppose that k≠0, k′≠0, and that k′ is conjugate to k. Then:

∑ab<(∑ak)1/k(∑bk′)1/k′

holds with (k>1), unless (ak) and (bk′) are proportional; and

∑ab>(∑ak)1/k(∑bk′)1/k′

holds with (k<1), unless (ak) and (bk′) are proportional or (ab) is nul. Furthermore, those two inequalies can be combined in a single inequality:

(∑ab)kk′<(∑ak)k′(bk′)k

with k≠0 and k≠1.

General property of mean values:

§ If r<s then

¯r(a)<¯s(a)

, unless the a are all equal, or s≤0 and ana a is zero.