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 ≥ ... ≥ a_n and b₁ ≥ b₂ ≥ ... ≥ b_n,

Minkowski's Inequality

For positive a_k, b_k 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_k,

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 a_ij and transpose A^T. Then

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