If p is a non-zero real number, and are positive real numbers, then the generalized mean or power mean with exponent p of these positive real numbers is:
(See p-norm). For p = 0 we set it equal to the geometric mean

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Let be a sequence of positive real numbers, then the following properties hold:
1. .Each

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We will prove weighted power means inequality, for the purpose of the proof we will assume the following without loss of generality:

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The power mean could be generalized further to the generalized f-mean:
This covers the geometric mean without using a limit with f(x) = log(x). The power mean is obtained for f(x) =

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• Arithmetic–geometric mean
• Average
• Heronian mean
• Inequality of arithmetic and geometric means

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1. ^ Sýkora, Stanislav (2009). Mathematical means and averages: basic properties. 3. Stan’s Library: Castano Primo, Italy. doi:10.3247/SL3Math09.001.
2. ^ P. S. Bullen: Handbook of Means and

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Signal processing
A power mean serves a non-linear moving averagewhich is shifted towards small signal values for small p

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