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  "body": "\n## Statement\n\nLet $x_1, \\dots, x_n \u003e 0$ be real numbers and let $w_1, \\dots, w_n \u003e 0$ be weights with $w_1 + \\dots + w_n = 1$.\nFor a real number $p \\ne 0$, let\n$$M_p = \\Big( \\sum_{i=1}^{n} w_i x_i^{p} \\Big)^{1/p}, \\qquad M_0 = \\prod_{i=1}^{n} x_i^{w_i}.$$\nIf $p \u003c q$, then $M_p \\le M_q$, with equality if and only if $x_1 = x_2 = \\dots = x_n$.\n\n## Remarks\n\nProof idea: for $0 \u003c p \u003c q$, the function $t \\mapsto t^{q/p}$ is strictly convex on $(0, \\infty)$, and [[Jensen's inequality|0RWMRTPC87]] applied to the numbers $x_i^p$ gives $M_p^q \\le M_q^q$.\nThe case $p \u003c q \u003c 0$ follows by applying this to the numbers $1 / x_i$, since $M_{-r}(x) = 1 / M_r(1/x)$.\nThe cases with $p = 0$ or $q = 0$ follow from the weighted [[AM–GM inequality|0DRF0F3XC9]], or from the limit $M_p \\to M_0$ as $p \\to 0$.\n\nExample: the cases $p = -1, 0, 1, 2$ give harmonic mean $\\le$ geometric mean $\\le$ arithmetic mean $\\le$ quadratic mean.\nFor $x = (1, 4)$ with equal weights these are $8/5$, $2$, $5/2$ and $\\sqrt{17/2} \\approx 2.92$.\n\nMathlib states a related form as `Real.rpow_arith_mean_le_arith_mean_rpow`.\n\n## Explanations\n\n### L1 human\n\nThere are many ways to average positive numbers: the usual average, the geometric mean, the root mean square, and more.\nThey form one family indexed by an exponent $p$.\nThe inequality says that a larger exponent always gives a larger average, unless all the numbers are equal.\n\n### L2 human\n\nThe power mean inequality packages the AM–GM inequality, the AM–HM inequality and the AM–QM inequality into one statement.\nIt is a direct consequence of Jensen's inequality for the power functions.\nAs $p \\to \\infty$ the mean $M_p$ tends to $\\max_i x_i$, and as $p \\to -\\infty$ it tends to $\\min_i x_i$, so all the means lie between the smallest and the largest number.\n",
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