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  "body": "\n## Definition\n\nLet $I$ be a real [[interval|0Q1D85B8G6]] and let $\\varphi \\colon I \\to \\R$ be a [[continuous|0X3WGX95FE]] and [[strictly monotone|011EJQWWJS]] function.\nLet $n \\ge 1$ and let $x_1, \\dots, x_n \\in I$.\nThe *quasi-arithmetic mean* of $x_1, \\dots, x_n$ generated by $\\varphi$ is\n$$M_\\varphi(x_1, \\dots, x_n) = \\varphi^{-1}\\left( \\frac{1}{n} \\sum_{i=1}^{n} \\varphi(x_i) \\right),$$\nwhere $\\varphi^{-1}$ is the [[inverse function|0J2WAR83WP]] of $\\varphi$ on the interval $\\varphi(I)$.\n\n## Remarks\n\nThe value is well defined: the average of the $\\varphi(x_i)$ lies between their minimum and maximum, hence in the interval $\\varphi(I)$.\nExample: 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.\nNon-example: $\\varphi(x) = x^2$ on $I = \\R$ is not strictly monotone, and the formula breaks down.\nFor the numbers $-1$ and $1$ the value $\\varphi^{-1}(1)$ is not defined, since both $-1$ and $1$ map to $1$.\nOther 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).\nTwo functions $\\varphi$ and $\\psi$ generate the same mean exactly when $\\psi = a \\varphi + b$ with real $a \\ne 0$ and $b$.\nThe weighted version replaces $\\frac{1}{n}$ by weights $w_i \\ge 0$ with $\\sum_i w_i = 1$.\nThe mean is also called the Kolmogorov mean or the generalized $f$-mean.\n",
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