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18个基本不等式

时间:2012-09-29


A

Classical Inequalities

Theorem 1. (AM-GM inequality) Let a1 , · · · , an be positive real numbers. Then, we have √ a1 + · · · + an ≥ n a1 · · · an . n Theorem 2. (Weighted AM-GM inequality) Let λ1 , · · · , λn real positive numbers with λ1 +· · ·+λn = 1. For all x1 , · · · , xn > 0, we have λ1 · x1 + · · · + λn · xn ≥ x1 λ1 · · · xn λn . Theorem 3. (GM-HM inequality) Let a1 , · · · , an be positive real numbers. Then, we have √ n n a1 · · · an ≥ 1 1 + a2 + · · · + a1 a1 n Theorem 4. (QM-AM inequality) Let a1 , · · · , an be positive real numbers. Then, we have a2 + a2 + · · · + a2 a1 + · · · + an n 1 2 ≥ n n Theorem 5. (Power Mean inequality) Let x1 , · · · , xn > 0. The power mean of order p is de?ned by √ M0 (x1 , x2 , . . . , xn ) = n x1 · · · xn , Mp (x1 , x2 , . . . , xn ) = xp + · · · + xn p 1 n
1 p

(p = 0).

Then the function Mp (x1 , x2 , . . . , xn ) : R → R is continuous and monotone increasing.

Theorem 6. (Rearrangement inequality) Let x1 ≥ · · · ≥ xn and y1 ≥ · · · ≥ yn be real numbers. For any permutation σ of {1, . . . , n}, we have
n n n

xi yi ≥
i=1 i=1

xi yσ(i) ≥
i=1

xi yn+1?i .

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Theorem 7. (The Cauchy3 -Schwarz4 -Bunyakovsky5 inequality) Let a1 , · · · , an , b1 , · · · , bn be real numbers. Then, (a1 2 + · · · + an 2 )(b1 2 + · · · + bn 2 ) ≥ (a1 b1 + · · · + an bn )2 . Remark. This inequality apparently was ?rstly mentioned in a work of A.L. Cauchy in 1821. The integral form was obtained in 1859 by V.Y. Bunyakovsky. The corresponding version for inner-product spaces obtained by H.A. Schwartz in 1885 is mainly known as Schwarz’s inequality. In light of the clear historical precedence of Bunyakovsky’s work over that of Schwartz, the common practice of referring to this inequality as CS-inequality may seem unfair. Nevertheless in a lot of modern books the inequality is named CSB-inequality so that both Bunyakovsky and Schwartz appear in the name of this fundamental inequality. By setting ai =
x √i yi

and bi =



yi the CSB inequality takes the following form

Theorem 8. (Cauchy’s inequality in Engel’s form) Let x1 , · · · , xn , y1 , · · · , yn be positive real numbers. Then, x2 x2 x2 (x1 + x2 + · · · + xn ) 1 + 2 + ··· + n ≥ y1 y2 yn y1 + y2 + · · · + yn
2

Theorem 9. (Chebyshev’s inequality6 ) Let x1 ≥ · · · ≥ xn and y1 ≥ · · · ≥ yn be real numbers. We have x1 y1 + · · · + xn yn ≥ n x1 + · · · + xn n y1 + · · · + yn n .

Theorem 10. (H¨lder’s inequality7 ) o Let x1 , · · · , xn , y1 , · · · , yn be positive real numbers. Suppose that p > 1 and 1 q > 1 satisfy p + 1 = 1. Then, we have q
n n
1 p

n

1 q

xi yi ≤
i=1 i=1

xi p
i=1

yi q

3 Louis

Augustin Cauchy (1789-1857), french mathematician Amandus Schwarz (1843-1921), german mathematician 5 Viktor Yakovlevich Bunyakovsky (1804-1889), russian mathematician 6 Pafnuty Lvovich Chebyshev (1821-1894), russian mathematician. 7 Otto Ludwig H¨lder (1859-1937), german mathematician o
4 Hermann

150

Theorem 11. (Minkowski’s inequality8 ) If x1 , · · · , xn , y1 , · · · , yn > 0 and p > 1, then
n
1 p

n

1 p

n

1 p

xi p
i=1

+
i=1

yi p


i=1

(xi + yi )

p

De?nition 1. (Convex functions.) We say that a function f (x) is convex on a segment [a, b] if for all x1 , x2 ∈ [a, b] f x1 + x2 2 ≤ f (x1 ) + f (x2 ) 2

Theorem 12. (Jensen’s inequality9 ) Let n ≥ 2 and λ1 , . . . , λn be nonnegative real numbers such that λ1 +· · ·+λn = 1. If f (x) is convex on [a, b] then f (λ1 x1 + · · · + λn xn ) ≤ λ1 f (x1 ) + · · · + λn xn for all x1 , . . . , xn ∈ [a, b].

De?nition 2. (Majorization relation for ?nite sequences) Let a = (a1 , a2 , . . . , an ) and b = (b1 , b2 , . . . , bn ) be two (?nite) sequences of real numbers such that a1 ≥ a2 ≥ · · · ≥ an and b1 ≥ b2 ≥ · · · ≥ bn . We say that the sequence a majorizes the sequence b and we write a b or b a

if the following two conditions are satisfyied (i) a1 + a2 + · · · + ak ≥ b1 + b2 + · · · + bk , for all k, 1 ≤ k ≤ n ? 1; (ii) a1 + a2 + · · · + an = b1 + b2 + · · · + bn .

Theorem 13. (Majorization inequality | Karamata’s inequality10 ) Let f : [a, b] ?→ R be a convex function. Suppose that (x1 , · · · , xn ) majorizes (y1 , · · · , yn ), where x1 , · · · , xn , y1 , · · · , yn ∈ [a, b]. Then, we obtain f (x1 ) + · · · + f (xn ) ≥ f (y1 ) + · · · + f (yn ).
Minkowski (1864-1909), german mathematician. Ludwig William Valdemar Jensen (1859-1925), danish mathematician. 10 Jovan Karamata (1902-2967), serbian mathematician.
9 Johan 8 Hermann

151

Theorem 14. (Muirhead’s inequality11 | Bunching Principle ) If a = (a1 , a2 , . . . , an ) and b = (b1 , b2 , . . . , bn ) are two nonincreasing sequences of nonnegative real numbers such that a majorizes b, then we have xa1 · · · xan ≥ n 1
sym sym

xb1 · · · xbn n 1

where the sums are taken over all n! permutations of variables x1 , x2 , . . . , xn .

Theorem 15. (Schur’s inequality12 ) Let x, y, z be nonnegative real numbers. For any r > 0, we have xr (x ? y)(x ? z) ≥ 0.
cyc

Remark. The case r = 1 of Schur’s inequality is x3 ? 2x2 y + xyz ≥ 0
sym

By espanding both the sides and rearranging terms, each of following inequalities is equivalent to the r = 1 case of Schur’s inequality ? x3 + y 3 + z 3 + 3xyz ≥ xy(x + y) + yz(y + z) + zx(z + x) ? xyx ≥ (x + y ? z)(y + z ? x)(z + x ? y) ? (x + y + z)3 + 9xyz ≥ 4(x + y + z)(xy + yz + zx)

Theorem 16. (Bernoulli’s inequality13 ) For all r ≥ 1 and x ≥ ?1, we have (1 + x)r ≥ 1 + rx.

De?nition 3. (Symmetric Means) For given arbitrary real numbers x1 , · · · , xn , the coe?cient of tn?i in the polynomial (t + x1 ) · · · (t + xn ) is called the i-th elementary symmetric function σi . This means that (t + x1 ) · · · (t + xn ) = σ0 tn + σ1 tn?1 + · · · + σn?1 t + σn .
Muirhead (1860-1941), english matematician. Schur (1875-1941), was Jewish a mathematician who worked in Germany for most of his life. He considered himself German rather than Jewish, even though he had been born in the Russian Empire in what is now Belarus, and brought up partly in Latvia. 13 Jacob Bernouilli (1654-1705), swiss mathematician founded this inequality in 1689. However the same result was exploited in 1670 by the english mathematician Isaac Barrow.
12 Issai 11 Robert

152

For i ∈ {0, 1, · · · , n}, the i-th elementary symmetric mean Si is de?ned by Si = σi
n i

.

Theorem 17. (Newton’s inequality14 ) Let x1 , . . . , xn > 0. For i ∈ {1, · · · , n}, we have
2 Si ≥ Si?1 · Si+1

Theorem 18. (Maclaurin’s inequality15 ) Let x1 , . . . , xn > 0. For i ∈ {1, · · · , n}, we have S1 ≥ S2 ≥
3

S3 ≥ · · · ≥

n

Sn

14 Sir Isaac Newton (1643-1727), was the greatest English mathematician of his generation. He laid the foundation for di?erential and integral calculus. His work on optics and gravitation make him one of the greatest scientists the world has known. 15 Colin Maclaurin (1698-1746), Scottish mathematican.

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