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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.
For a1 ≥ a2 ≥ ... ≥ an and b1 ≥ b2 ≥ ... ≥ bn,
For positive ak, bk and p > 1,
For x > -1, x ≠ 0, and integers n > 1,
A special case is
For 0 < p ≤ q and positive ak,
Cauchy-Schwarz Inequality for Integrals
Hölder's Inequality for Integrals
Minkowski's Inequality for Integrals
For p > 1,
Jensen's Inequality for Integrals
For 0 < p < q,
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 aij and transpose AT. Then
Last edited by Zhylliolom (2006-08-06 18:35:47)
It may be much better when equal conditions are given too.