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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
akbk (  n
ak )(  n
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

( ≤ e  

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
kxk) )  n

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.

H�lder'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 (mirror)

Mathworld: H�lder'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


Jensen's inequality can be used to solve the "sum of sqrts" puzzle.

The Quadratic Formula

The webmaster and author of this Math Help site is Graeme McRae.