general means inequality

If 0r, the r-mean (or r-th power meanMathworldPlanetmath) of the nonnegative numbers a1,,an is defined as


Given real numbers x,y such that xy0 and x<y, we have


and the equality holds if and only if a1==an.

Additionally, if we define M0 to be the geometric meanMathworldPlanetmath (a1a2an)1/n, we have that the inequalityMathworldPlanetmath above holds for arbitrary real numbers x<y.

The mentioned inequality is a special case of this one, since M1 is the arithmetic meanMathworldPlanetmath, M0 is the geometric mean and M-1 is the harmonic meanMathworldPlanetmath.

This inequality can be further generalized using weighted power means.

Title general means inequality
Canonical name GeneralMeansInequality
Date of creation 2013-03-22 12:39:49
Last modified on 2013-03-22 12:39:49
Owner drini (3)
Last modified by drini (3)
Numerical id 6
Author drini (3)
Entry type Theorem
Classification msc 26D15
Synonym power means inequality
Related topic ArithmeticGeometricMeansInequality
Related topic ArithmeticMean
Related topic GeometricMean
Related topic HarmonicMean
Related topic PowerMean
Related topic ProofOfArithmeticGeometricHarmonicMeansI
Related topic RootMeanSquare3
Related topic DerivationOfZerothWeightedPowerMean
Related topic ProofOfArithmeticGeometricHarmonicMeansInequality
Related topic ComparisonOfPythagor