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**

**Hadamard's Inequality**

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

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