Famous Inequalities
Misc Inequalities
Jordan's Inequality
(2/π)x ≤ sin(x) ≤ x for all x in [0,π/2]
Proof: sin's slope is positive (but never greater than 1) in the interval,
and concave down, so the graphs of
y=x, y=sin(x), y=(2/π)x touch only at the
endpoints of the interval.
Young's inequality
Given f, a strictly increasing function whose value is 0 at 0, the
integral from 0 to a of f plus the integral from 0 to b of f-1 is
greater than the product ab. This is fairly obvious from the area
interpretation of the integral, but not so obvious is that choosing f(x)=xp-1
results in this famous special case of Young's Inequality:
Let a, b > 0; p, q > 0; 1/p + 1/q = 1. Then ab ≤ ap/p + bq/q
Proof: log is concave down, so log ab = (1/p)log ap + (1/q)log bq
≤ log (1/p)ap + log (1/q)bq
Bernoulli's inequality
Let x > -1, r > 1. Then (1+x)r ≥ 1+xr
Nesbitt's inequality
For a,b,c>0, a/(b+c)+b/(a+c)+c/(a+b) ≥ 3/2
Shapiro inequality
For x1,x2,...,xn > 0, n ≤ 12 or n is odd and n
≤ 23, then
the sum xi/(xi+1+xi+2) ≥ n/2, where the subscripts are understood modulo n.
Schur's Inequality
For all non-negative real numbers x, y, z and a positive number t,
xt(x-y)(x-z) + yt(y-z)(y-x) + zt(z-x)(z-y) ≥ 0
Inequalities for sequences
Chebyshev's inequality
If a1 ≥ a2 ≥ ... ≥ an and b1
≥ b2 ≥ ... ≥ bn
then
| n |
n
Σ
k=1 |
akbk ≥ |
( |
n
Σ
k=1 |
ak |
)( |
n
Σ
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 the LHS, so the LHS is at least as big
as the RHS.
MacLaurin's Inequality
MacLaurin's inequality is a generalization of the AM-GM inequality.
First, define the product of a set as the product of all the elements of
the set.
Next, define the MacLaurin Sum Sk as the average of all the
products of k-element subsets of an n-element set.
The MacLaurin inequality is S1 ≥ S21/2
≥ S31/3 ≥ ... ≥ Sn1/n
Thus the AM-GM inequality represents the endpoints of a longer chain of
inequalities.
Carleman's inequality
Let a1, a2, ... be a sequence of real
numbers. Carleman's inequality states that
∞
Σ
n=1 |
(a1a2...an)1/n ≤ e |
∞
Σ
n=1 |
an |
Arithmetic-geometric-harmonic means inequality
Let x1, x2, ..., xn be positive
numbers. Then
Max(x1, x2, ..., xn) ≥
(x1 + x2 + ... + xn)/n (the
arithmetic mean) ≥
(x1 x2 ... xn)1/n (the
geometric mean) ≥
n/(1/x1 + 1/x2 + ... + 1/xn) (the harmonic
mean) ≥
Min(x1, x2, ..., xn),
(with equality only if x1=x2=...=xn)
General Means inequality, a.k.a. Power Means inequality
The power means inequality is a generalization of the AM-GM-HM
inequality.
Define Mr, the rth power mean of nonnegative
numbers a1, a2, ... an as
When r ≠ 0, Mr(a1,
a2, ... an) = Average(a1r, a2r,
... anr)1/r
The limit of Mr, as r approaches zero, is the Geometric Mean,
so
when r=0, Mr(a1, a2, ... an) = (a1
a2 ... an)1/n, the Geometric
Mean.
The Power Means inequality states that whenever x<y, Mx(a1,
a2, ... an) ≤ My(a1,
a2, ... an)
(with equality only if a1=a2=...=an)
Jensen's inequality
If f is a convex function, then the value of the function at the weighted
average of selected values of its domain is less than or equal to the
weighted average (same weights) of the values of the function of those same
selected values of its domain.
| f |
( |
|
n
Σ
k=1 |
(λkxk) |
) |
≤ |
n
Σ
k=1 |
λkf(xk) |
If f is a concave function (sometimes called convex down) then the
inequality is reversed.
Minkowski inequality
Rearrangement inequality
If x1, x2, ..., xn and y1, y2,
..., yn are two sequences of positive numbers, then the sum
x1y1 + x2y2 + ... + xnyn
is maximized when the two sequences are ordered the same way (i.e. x1
≤ x2 ≤ ... ≤ xn and y1 ≤
y2 ≤ ... ≤ yn) and is minimized when the two sequences are ordered opposite
to one another.
Hölder's Inequality
Let a1, ..., an; b1, ..., bn;
... ; z1, ..., zn be sequences of nonnegative real
numbers, and let la, lb,
..., lz be positive reals which sum to
1. Then:
(a1 + ... + an)la(b1
+ ... + bn)lb...(z1
+ ... + zn)lz
≥ a1lab1lb...z1lz
+ ... + anlabnlb...znlz
Cauchy's inequality is a special case of this in which n=2 and la
= lb = 1/2.
Geometric inequalities
Hadwiger-Finsler inequality
Weizenbock's Inequality
Brunn-Minkowski inequality
Internet References
Wikipedia: List
of Inequalities, Schur's
inequality
PlanetMath.Org, Inequalities
for Real Numbers
Topics in
Inequalities, by Hojoo Lee
Mathworld: Hölder's
Inequality
Related Pages in this website
Go back to the Number Theory Home
The AM-GM
Inequality: the Arithmetic Mean of positive numbers is always greater than
the Geometric Mean. This is proved using Jensen's Inequality.
The Triangle Inequality
The Chebyshev Sum
Inequality
Puzzles
Jensen's inequality can be used to solve the
"sum of sqrts" puzzle.
The Quadratic Formula
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