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

$\overline{x}(m)={(\frac{1}{n}\cdot \sum _{i=1}^n|x_i{|}^m)}^{1/m}$

and ordered this way:

$\begin{array}{cccc}& 𝑚𝑖𝑛𝑖𝑚𝑢𝑚& =& \overline{x}(-\mathrm{\infty })\\ <& ℎ𝑎𝑟𝑚𝑜𝑛𝑖𝑐𝑚𝑒𝑎𝑛& =& \overline{x}(-1)\\ <& 𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐𝑚𝑒𝑎𝑛& =& \underset{m\to 0}{\lim }\overline{x}(m)\\ <& 𝑎𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐𝑚𝑒𝑎𝑛& =& \overline{x}(1)\\ <& 𝑚𝑎𝑥𝑖𝑚𝑢𝑚& =& \overline{x}(\mathrm{\infty })\end{array}$