proof of arithmetic-geometric-harmonic means inequality

We can use the Jensen inequalityMathworldPlanetmath for an easy proof of the arithmetic-geometric-harmonic means inequality.

Let x1,,xn>0; we shall first prove that


Note that log is a concave functionMathworldPlanetmath. Applying it to the arithmetic meanMathworldPlanetmath of x1,,xn and using Jensen’s inequalityMathworldPlanetmath, we see that

log(x1++xnn) log(x1)++log(xn)n

Since log is also a monotone function, it follows that the arithmetic mean is at least as large as the geometric meanMathworldPlanetmath.

The proof that the geometric mean is at least as large as the harmonic meanMathworldPlanetmath is the usual one (see “proof of arithmetic-geometric-harmonic means inequality”).

Title proof of arithmetic-geometric-harmonic means inequality
Canonical name ProofOfArithmeticgeometricharmonicMeansInequality
Date of creation 2013-03-22 12:43:07
Last modified on 2013-03-22 12:43:07
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Example
Classification msc 39B62
Classification msc 26D15
Related topic ArithmeticGeometricMeansInequality
Related topic ProofOfArithmeticGeometricMeansInequalityUsingLagrangeMultipliers