Editing Quasi-arithmetic mean (section)

== Examples ==

* If <math>I = \mathbb{R}</math>, the [[real line]],  and <math>f(x) = x</math>, (or indeed any linear function <math>x\mapsto a\cdot x + b</math>, <math>a</math> not equal to 0) then the ''f''-mean corresponds to the [[arithmetic mean]].
* If <math>I = \mathbb{R}^+</math>, the [[positive real numbers]] and <math>f(x) = \log(x)</math>, then the ''f''-mean corresponds to the [[geometric mean]]. According to the ''f''-mean properties, the result does not depend on the base of the [[logarithm]] as long as it is positive and not 1.
* If <math>I = \mathbb{R}^+</math> and <math>f(x) = \frac{1}{x}</math>, then the ''f''-mean corresponds to the [[harmonic mean]].
* If <math>I = \mathbb{R}^+</math> and <math>f(x) = x^p</math>, then the ''f''-mean corresponds to the [[power mean]] with exponent <math>p</math>.
* If <math>I = \mathbb{R}</math> and <math>f(x) = \exp(x)</math>, then the ''f''-mean is the mean in the [[log semiring]], which is a constant shifted version of the [[LogSumExp]] (LSE) function (which is the logarithmic sum), <math>M_f(x_1, \dots, x_n) = \mathrm{LSE}(x_1, \dots, x_n)-\log(n)</math>. The <math>-\log(n)</math> corresponds to dividing by {{mvar|''n''}}, since logarithmic division is linear subtraction. The LogSumExp function is a [[smooth maximum]]: a smooth approximation to the maximum function.