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The weighted arithmetic-geometric mean inequality for real weights

Statement

Let $n\ge1$ , let $x_1,\ldots,x_n>0$ , and let $w_1,\ldots,w_n\ge0$ satisfy $\sum_{i=1}^nw_i=1$ . Then $\prod_{i=1}^n x_i^{w_i}\le\sum_{i=1}^nw_ix_i.$

Facts & Assumptions

Given: Positive reals $x_i$ and nonnegative real weights $w_i$ summing to $1$ .

[L1]

The two-point exponential inequality holds for every weight in $[0,1]$ (The two-point convexity inequality for the exponential function).

[L2]

Positive-base real powers and their product laws are $x^r=\exp(r\log x)$ and the laws of The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents (Real powers for positive bases, with the zero-base positive-exponent convention, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm).

[L3]

Mathematical induction is valid on natural numbers (The principle of mathematical induction).

Proof

technique · induction

1.1

For $n=1$ , $w_1=1$ and both sides are $x_1$ .

basegiven

1.2

Assume the result for $n$ positive entries. For weights $w_1,\ldots,w_{n+1}$ , if $w_{n+1}=1$ the claim is immediate; otherwise put $W=1-w_{n+1}>0$ , $v_i=w_i/W$ for $i\le n$ , and $P:=\prod_{i=1}^nx_i^{v_i}$ .

ihgiven

1.3

Applying [L1] to $\log P,\log x_{n+1}$ with weights $W,w_{n+1}$ gives $P^Wx_{n+1}^{w_{n+1}}\le WP+w_{n+1}x_{n+1}$ .

L1L2

2.1

The $v_i$ are nonnegative and sum to one, so the induction hypothesis gives $P\le A:=\sum_{i=1}^nv_ix_i$ .

step 1.2ih

3.1

The left side in step 1.3 is $\prod_{i=1}^{n+1}x_i^{w_i}$ , and step 2.1 makes its right side at most $\sum_{i=1}^{n+1}w_ix_i$ .

step 2.1step 1.3L2algebra

4.1

The base and induction steps prove the inequality for every $n\ge1$ .

step 1.1step 3.1L3discharge-induction∎

Depends on

Used by

Young's inequality for conjugate real exponents Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 60 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)