Generalized mean

In mathematics, the Hölder means of the Hölder mean ( after Otto Hölder, 1859-1937 ) or the Potency (English and others (p- th ) power mean) is a (sometimes the ) generalized mean. The term is not uniform, designations such as the - te, means the order or the degree or exponent are also in circulation. In English, it is also referred to as a generalized mean.

Similarly, inconsistencies in the level of writing, instead of being well, or written.

The Hölder means generalizes the well-known since the Pythagoreans averages such as the arithmetic, geometric, square and harmonic mean by introducing a parameter

• 3.1 Weighted Hölder means
• 3.2 f - agent

Definition

Holder for a real number, the agent of the numbers is defined as a step for

The root notation is usually used only for natural numbers.

A matching definition for

Properties

• The Hölder means is homogeneous with respect to, ie

• Also, applies

• An important inequality of the Hölder means

• The Potency values ​​are consistent with the sample moments about zero quite simple relationship:

• In the stochastic convergence is defined in the pth mean of these Potency values ​​.

Special cases

Means choosing a suitable parameter yields the known mean values ​​:

Further generalizations

Weighted Hölder means

Also to the Hölder means can define a Weighted means: The weighted Hölder means can be defined with the weights with as

Being used for the "normal" Hölder means.

F means

The Hölder means can be further generalized to

This is a function of; the Hölder agent.