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)=x^{p1}
results in this famous special case of Young's Inequality:
Let a, b > 0; p, q > 0; 1/p + 1/q = 1. Then ab ≤ a^{p}/p +
b^{q}/q
Proof: log is concave down, so log ab = (1/p)log a^{p} + (1/q)log b^{q}
≤ log (1/p)a^{p} + log (1/q)b^{q}
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 x_{1},x_{2},...,x_{n} > 0, n ≤ 12 or n is odd
and n ≤ 23, then
the sum x_{i}/(x_{i+1}+x_{i+2}) ≥ n/2, where the
subscripts are understood modulo n.
Schur's Inequality
For all nonnegative real numbers x, y, z and a positive number t,
x^{t}(xy)(xz) + y^{t}(yz)(yx) + z^{t}(zx)(zy)
≥ 0
Chebyshev's inequality
If a_{1} ≥ a_{2} ≥ ... ≥ a_{n} and b_{1}
≥ b_{2} ≥ ... ≥ b_{n}
then
n 
_{ n}
Σ
^{
k=1} 
a_{k}b_{k} ≥ 
( 
_{ n}
Σ
^{
k=1} 
a_{k} 
)( 
_{ n}
Σ
^{
k=1} 
b_{k} 
) 
The proof is that the RHS can be written as the sum of n different
sums,
(a_{1}b_{1} + a_{2}b_{2} + ...
+ a_{n1}b_{n1} + a_{n}b_{n}) +
(a_{1}b_{2} + a_{2}b_{3} + ...
+ a_{n1}b_{n} + a_{n}b_{1}) +
...
(a_{1}b_{n} + a_{2}b_{1} + ...
+ a_{n1}b_{n2} + a_{n}b_{n1})
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 AMGM inequality.
First, define the product of a set as the product of all the elements of
the set.
Next, define the MacLaurin Sum S_{k} as the average of all the
products of kelement subsets of an nelement set.
The MacLaurin inequality is S_{1} ≥ S_{2}^{1/2}
≥ S_{3}^{1/3} ≥ ... ≥ S_{n}^{1/n}
Thus the AMGM inequality represents the endpoints of a longer chain of
inequalities.
Carleman's inequality
Let a_{1}, a_{2}, ... be a sequence of real numbers.
Carleman's inequality states that
_{ }∞_{
}
Σ
^{n=1} 
(a_{1}a_{2}...a_{n})^{1/n} ≤ e 
_{ }∞_{
}
Σ
^{n=1} 
a_{n} 
Arithmeticgeometricharmonic means inequality
Let x_{1}, x_{2}, ..., x_{n} be positive numbers.
Then
Max(x_{1}, x_{2}, ..., x_{n}) ≥
(x_{1} + x_{2} + ... + x_{n})/n (the
arithmetic mean) ≥
(x_{1} x_{2} ... x_{n})^{1/n} (the
geometric mean) ≥
n/(1/x_{1} + 1/x_{2} + ... + 1/x_{n}) (the harmonic
mean) ≥
Min(x_{1}, x_{2}, ..., x_{n}),
(with equality only if x_{1}=x_{2}=...=x_{n})
General Means inequality, a.k.a. Power Means inequality
The power means inequality is a generalization of the AMGMHM
inequality.
Define M^{r}, the r^{th} power mean of nonnegative
numbers a_{1}, a_{2}, ... a_{n} as
When r ≠ 0, M^{r}(a_{1}, a_{2}, ... a_{n})
= Average(a_{1}^{r}, a_{2}^{r}, ... a_{n}^{r})^{1/r}
The limit of M^{r}, as r approaches zero, is the Geometric Mean,
so
when r=0, M^{r}(a_{1}, a_{2}, ... a_{n}) =
(a_{1}
a_{2} ... a_{n})^{1/n}, the Geometric Mean.
The Power Means inequality states that whenever x<y, M^{x}(a_{1},
a_{2}, ... a_{n}) ≤ M^{y}(a_{1}, a_{2},
... a_{n})
(with equality only if a_{1}=a_{2}=...=a_{n})
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} 
(λ_{k}x_{k}) 
) 
≤ 
_{ n
}
Σ
^{k=1} 
λ_{k}f(x_{k}) 
If f is a concave function (sometimes called convex down) then the
inequality is reversed.
Minkowski inequality
Rearrangement inequality
If x_{1}, x_{2}, ..., x_{n} and y_{1}, y_{2},
..., y_{n} are two sequences of positive numbers, then the sum
x_{1}y_{1} + x_{2}y_{2} + ... + x_{n}y_{n}
is maximized when the two sequences are ordered the same way (i.e. x_{1}
≤ x_{2} ≤ ... ≤ x_{n} and y_{1} ≤ y_{2} ≤
... ≤ y_{n}) and is minimized when the two sequences are ordered
opposite to one another.
H�lder's Inequality
Let a_{1}, ..., a_{n}; b_{1}, ..., b_{n};
... ; z_{1}, ..., z_{n} be sequences of nonnegative real
numbers, and let l_{a},
l_{b}, ..., l_{z} be
positive reals which sum to 1. Then:
(a_{1} + ... + a_{n})^{l}^{a}(b_{1}
+ ... + b_{n})^{l}^{b}...(z_{1}
+ ... + z_{n})^{l}^{z}
≥ a_{1}^{l}^{a}b_{1}^{l}^{b}...z_{1}^{l}^{z}
+ ... + a_{n}^{l}^{a}b_{n}^{l}^{b}...z_{n}^{l}^{z}
Cauchy's inequality is a special case of this in which n=2 and
l_{a}
= l_{b} = 1/2.