Real functions Classical inequalities Theorem

Generalized mean inequality

Builds on Jensen's inequality Inequality of arithmetic and geometric means

Statement

Let $x_1, \dots, x_n > 0$ be real numbers and let $w_1, \dots, w_n > 0$ be weights with $w_1 + \dots + w_n = 1$ . For a real number $p \ne 0$ , let $M_p = \Big( \sum_{i=1}^{n} w_i x_i^{p} \Big)^{1/p}, \qquad M_0 = \prod_{i=1}^{n} x_i^{w_i}.$ If $p < q$ , then $M_p \le M_q$ , with equality if and only if $x_1 = x_2 = \dots = x_n$ .

Remarks

Proof idea: for $0 < p < q$ , the function $t \mapsto t^{q/p}$ is strictly convex on $(0, \infty)$ , and Jensen's inequality applied to the numbers $x_i^p$ gives $M_p^q \le M_q^q$ . The case $p < q < 0$ follows by applying this to the numbers $1 / x_i$ , since $M_{-r}(x) = 1 / M_r(1/x)$ . The cases with $p = 0$ or $q = 0$ follow from the weighted AM–GM inequality, or from the limit $M_p \to M_0$ as $p \to 0$ .

Example: the cases $p = -1, 0, 1, 2$ give harmonic mean $\le$ geometric mean $\le$ arithmetic mean $\le$ quadratic mean. For $x = (1, 4)$ with equal weights these are $8/5$ , $2$ , $5/2$ and $\sqrt{17/2} \approx 2.92$ .

Mathlib states a related form as Real.rpow_arith_mean_le_arith_mean_rpow.

Going deeper

For a student

The power mean inequality packages the AM–GM inequality, the AM–HM inequality and the AM–QM inequality into one statement. It is a direct consequence of Jensen's inequality for the power functions. As $p \to \infty$ 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.

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Convex function definition
Inequality of arithmetic and geometric means statement
Jensen's inequality statement
Generalized mean inequality statement

Tag	`XZ1EXQPJ6S`
URI	`m2://XZ1EXQPJ6S`
Kind	statement, theorem
Writing	W0 stub
MSC	26E60, 26D15
Status	This copy of the site has no build data. Run `m2 status XZ1EXQPJ6S`.
About	Jensen's inequality
Uses	Jensen's inequality, Inequality of arithmetic and geometric means
Changed	2 November 2026