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## #1 2006-08-05 20:31:33

Zhylliolom
Real Member
Registered: 2005-09-05
Posts: 412

### Inequalities

Triangle Inequality

or

Cauchy-Schwarz Inequality

Arithmetic, Geometric, and Harmonic Means

In other words, if A, G, and H are the arithmetic, geometric, and harmonic means, respectively, then H ≤ G ≤ A.

Hölder's Inequality

where

Chebyshev's Inequality

For a[sub]1[/sub] ≥ a[sub]2[/sub] ≥ ... ≥ a[sub]n[/sub] and b[sub]1[/sub] ≥ b[sub]2[/sub] ≥ ... ≥ b[sub]n[/sub],

Minkowski's Inequality

For positive a[sub]k[/sub], b[sub]k[/sub] and p > 1,

Bernoulli's Inequality

For x > -1, x ≠ 0, and integers n > 1,

A special case is

Jensen's Inequality

For 0 < p ≤ q and positive a[sub]k[/sub],

Cauchy-Schwarz Inequality for Integrals

Hölder's Inequality for Integrals

where

Minkowski's Inequality for Integrals

For p > 1,

Jensen's Inequality for Integrals

For 0 < p < q,

Young's Inequality

Let f be a continuous strictly increasing real-valued function on [0, ∞), with f(0) = 0 and as x approaches ∞, f(x) approaches ∞. Then if g is the inverse function of f, for any positive numbers a, b we have

Triangle Inequality for Vectors

Schwarz's Inequality for Vectors

Cauchy-Schwarz Inequality for Inner Product Spaces

Let A be an n × n matrix with entries a[sub]ij[/sub] and transpose A[sup]T[/sup]. Then

Last edited by Zhylliolom (2006-08-05 20:35:47)

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## #2 2006-09-30 19:15:15

George,Y
Member
Registered: 2006-03-12
Posts: 1,333

### Re: Inequalities

It may be much better when equal conditions are given too.

X'(y-Xβ)=0

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