Editing Quasi-arithmetic mean (section)

==Definition==

If ''f'' is a function which maps an interval <math>I</math> of the real line to the [[real number]]s, and is both [[continuous function|continuous]] and [[injective function|injective]], the '''''f''-mean of <math>n</math> numbers'''
<math>x_1, \dots, x_n \in I</math>
is defined as <math>M_f(x_1, \dots, x_n) = f^{-1}\left( \frac{f(x_1)+ \cdots + f(x_n)}n \right)</math>, which can also be written 
:<math> M_f(\vec x)= f^{-1}\left(\frac{1}{n} \sum_{k=1}^{n}f(x_k) \right)</math>

We require ''f'' to be injective in order for the [[inverse function]] <math>f^{-1}</math> to exist. Since <math>f</math> is defined over an interval, <math>\frac{f(x_1)+ \cdots + f(x_n)}n</math> lies within the domain of <math>f^{-1}</math>.

Since ''f'' is injective and continuous, it follows that ''f'' is a strictly [[monotonic function]], and therefore that the ''f''-mean is neither larger than the largest number of the tuple <math>x</math> nor smaller than the smallest number in <math>x</math>.