Quasi-arithmetic mean

Remarks

The value is well defined: the average of the $\varphi(x_i)$ lies between their minimum and maximum, hence in the interval $\varphi(I)$. Example: on $I = (0, \infty)$ with $\varphi = \ln$, $M_\varphi(x_1, \dots, x_n) = \exp\big(\frac{1}{n} \sum \ln x_i\big) = (x_1 \cdots x_n)^{1/n}$, the geometric mean. Non-example: $\varphi(x) = x^2$ on $I = \R$ is not strictly monotone, and the formula breaks down. For the numbers $-1$ and $1$ the value $\varphi^{-1}(1)$ is not defined, since both $-1$ and $1$ map to $1$. Other standard cases are $\varphi(x) = x$ (arithmetic mean), $\varphi(x) = 1/x$ (harmonic mean) and $\varphi(x) = x^p$ on $(0, \infty)$ with $p \ne 0$ (power mean). Two functions $\varphi$ and $\psi$ generate the same mean exactly when $\psi = a \varphi + b$ with real $a \ne 0$ and $b$. The weighted version replaces $\frac{1}{n}$ by weights $w_i \ge 0$ with $\sum_i w_i = 1$. The mean is also called the Kolmogorov mean or the generalized $f$-mean.