Weighted Power Mean

Weighted F-Mean (Mean in an Arbitrary Function)

Given any function which is both Continuous and Injective (So that is well defined), The f-mean of with weights is defined as

Since is both Continuous and Injective it follows that is strictly monotonic so it follows that .

Weighted Power Mean

The weighted power mean is the case of and of the f-mean.

Proof

Given with weights where and , The weighted power mean is

The weights serve to emphasises a certain element so if is more important then will be greater. In the case where , we get an unweighted/regular version. Some special case for are: . is valid for .

Weighted Geometric Mean

The weighted geometric mean is the case of the f-mean where and

Proof

2. Using logarithm laws we get that
3. So
4. W.L.O.G we can assume that to get a cleaner version and obtain

It is also the of the power mean Proof

4. indeterminate form we can use L’Hopitals Rule
6. Therefore

Weighted Power Mean Inequality

The equality holds if and only if . For the cases of we get the inequality . Again the equality only holding when all elements are equal.

Proof

1. let . Define then either:
   4. All of which is convex on .
2. Take our non-negative reals .
3. Then apply Jensen’s Inequality Inequality (Also ).
5. Raising both sides to the power gives as desired.
6. We’ve already established that is the geometric mean. Since is continuous and monotonic we can take the limit of the inequality to complete it

Young’s Inequality

Let . Then if

Where equality holds iff

Proof

1. This is just the case of AM-GM with