proof of general means inequality
Let , , …, be positive real numbers such that . For any real number , the weighted power mean of degree of positive real numbers , , …, (with respect to the weights , …, ) is defined as
The definition is extended to the case by taking the limit ; this yields the weighted geometric mean
(see derivation of zeroth weighted power mean). We will prove the weighted power means inequality, which states that for any two real numbers , the weighted power means of orders and of positive real numbers , , …, satisfy the inequality
with equality if and only if all the are equal.
First, let us suppose that and are nonzero. We distinguish three cases for the signs of and : , , and . Let us consider the last case, i.e. assume and are both positive; the others are similar. We write and for ; this implies . Consider the function
Since , the second derivative of satisfies for all , so is a strictly convex function. Therefore, according to Jensen’s inequality,
with equality if and only if . By substituting and back into this inequality, we get
with equality if and only if . Since is positive, the function is strictly increasing, so raising both sides to the power preserves the inequality:
which is the inequality we had to prove. Equality holds if and only if all the are equal.
If , the inequality is still correct: is defined as , and since for all with , the same holds for the limit . The same argument shows that the inequality also holds for , i.e. that for all . We conclude that for all real numbers and such that ,
Title | proof of general means inequality |
Canonical name | ProofOfGeneralMeansInequality |
Date of creation | 2013-03-22 13:10:26 |
Last modified on | 2013-03-22 13:10:26 |
Owner | pbruin (1001) |
Last modified by | pbruin (1001) |
Numerical id | 5 |
Author | pbruin (1001) |
Entry type | Proof |
Classification | msc 26D15 |
Related topic | ArithmeticMean |
Related topic | GeometricMean |
Related topic | HarmonicMean |
Related topic | RootMeanSquare3 |
Related topic | PowerMean |
Related topic | WeightedPowerMean |
Related topic | ArithmeticGeometricMeansInequality |
Related topic | JensensInequality |