proof of arithmetic-geometric-harmonic means inequality
We can use the Jensen inequality for an easy proof of the arithmetic-geometric-harmonic means inequality.
Let x1,…,xn>0; we shall first prove that
n√x1⋅…⋅xn≤x1+…+xnn. |
Note that log is a concave function. Applying it to the
arithmetic mean
of x1,…,xn and using Jensen’s inequality
, we see that
log(x1+…+xnn) | ≥log(x1)+…+log(xn)n | ||
=log(x1⋅…⋅xn)n | |||
=logn√x1⋅…⋅xn. |
Since log is also a monotone function, it follows that the arithmetic mean is at least as large as the geometric mean.
The proof that the geometric mean is at least as large as the harmonic mean 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 |