Statistics/Summary/Averages/Relationships among Arithmetic, Geometric and Harmonic Mean

Relationships among Arithmetic, Geometric and Harmonic Mean
The Means mentioned above are realizations of the generalized mean
 * $$ \bar{x}(m) = \left ( \frac{1}{n}\cdot\sum_{i=1}^n{|x_i|^m} \right ) ^{1/m} $$

and ordered this way:



\begin{alignat}{2} &\mathit{minimum}&\;=\;&\bar{x}(-\infty) \\ \mathrel<\;&\mathit{harmonic\ mean}&\;=\;&\bar{x}(-1) \\ \mathrel<\;&\mathit{geometric\ mean}&\;=\;&\lim_{m\rightarrow0}\bar{x}(m) \\ \mathrel<\;&\mathit{arithmetic\ mean}&\;=\;&\bar{x}(1) \\ \mathrel<\;&\mathit{maximum}&\;=\;&\bar{x}(\infty) \end{alignat} $$