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#1 2006-08-06 18:31:33

Zhylliolom
Real Member

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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 a1 ≥ a2 ≥ ... ≥ an and b1 ≥ b2 ≥ ... ≥ bn,



Minkowski's Inequality

For positive ak, bk 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 ak,



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



Hadamard's Inequality

Let A be an n × n matrix with entries aij and transpose AT. Then

Last edited by Zhylliolom (2006-08-06 18:35:47)

 

#2 2006-10-01 17:15:15

George,Y
Super Member

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Re: Inequalities

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


X'(y-Xβ)=0
 

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