Following chapter 2 of Greek Means and the arithmetic-geometric mean by George and Silvia Toader, the Greek (or Pythagorean) means of two numbers have been studied for centuries originating in the Pythagorean School. The Greek means labeled $m$ of two numbers $a$ and $b$ are defined by equating one of

$$\frac{a-m}{a-b},\space\frac{a-b}{m-b},\space\text{or}\space\frac{a-m}{m-b}$$ and $$\frac{a}{a},\space\frac{a}{b},\space\frac{b}{a},\space\frac{a}{m},\space\frac{m}{a},\space\frac{b}{m},\space\text{or}\space\frac{m}{b}$$

This yields 21 Greek means, but only 10 are unique and nontrivial.

This is a nice general definition for all means of two numbers, but what is a general definition for all means of $n$ numbers?


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