A.2 Means

Definition A..4 (Weighted Mean)   $p\in\mathbb{R}^n$ is called a vector of weights if $p_k>0$ for all $k\in\{1,\ldots,n\}$. For $\alpha\in\mathbb{R}$, $x\in\mathbb{R}^n$, and $p$ a vector of weights we define the weighted $\alpha$-mean of $x$ as follows:

1. $\alpha>0$: $M_\alpha(x,p) := \left( \sum p_k x_k^r / \sum p_k \right)^{(1/r)}$,
2. $\alpha=0$: $M_0(x,p) := \left( \prod x_k^{p_k} \right)^{1/\sum p_k}$,
3. $\alpha<0$ and one $x_k=0$: $M_\alpha(x,p):=0$.

In the case where all $p_k$ are equal we also write shorter $M_\alpha(x):=M_\alpha(x,p)$.

Theorem A..5 (Inequality of the Means)   If $\alpha<\beta$ then

$$\qquad M_\alpha(x) \le M_\beta(x) \qquad\forall x\in\mathbb{R}^n,$$

unless the $x_k$ are all equal, or $s\le0$ and one $x_k$ is zero.

Remark. Many more properties of means can be found in [43].

Remark. For $x\in\mathbb{R}^n$, obviously $M_2(x) = \sqrt{\sum x_k^2/n} = {1\over\sqrt{n}} \Vert x\Vert _2$.