Proofs of AM-GM

This pages lists some proofs of the weighted AM-GM Inequality. The inequality's statement is as follows: for all nonnegative reals $a_1, \dotsc, a_n$ and nonnegative reals $\lambda_1, \dotsc, \lambda_n$ such that $\sum_{i=1}^n \lambda_i = 1$ , then $\sum_{i=1}^n \lambda_i a_i \ge \prod_{i=1}^n a_i^{\lambda_i},$ with equality if and only if $a_i = a_j$ for all $i,j$ such that $\lambda_i, \lambda_j \neq 0$ .

We first note that we may disregard any $a_i$ for which $\lambda_i= 0$ , as they contribute to neither side of the desired inequality. We also note that if $a_i= 0$ and $\lambda_i \neq 0$ , for some $i$ , then the right-hand side of the inequality is zero and the right hand of the inequality is greater or equal to zero, with equality if and only if $a_j = 0 = a_i$ whenever $\lambda_j\neq 0$ . Thus we may henceforth assume that all $a_i$ and $\lambda_i$ are strictly positive.

Complete Proofs

Proof by Convexity

We note that the function $x \mapsto \ln x$ is trictly concave. Then by Jensen's Inequality, $\ln \sum_i \lambda_i a_i \ge \lambda_i \sum_i \ln a_i = ln \prod_i a_i^{\lambda_i} ,$ with equality if and only if all the $a_i$ are equal. Since $x \mapsto \ln x$ is a strictly increasing function, it then follows that $\sum_i \lambda_i a_i \ge \prod_i a_i^{\lambda_i},$ with equality if and only if all the $a_i$ are equal, as desired. $\blacksquare$

Proofs of Unweighted AM-GM

These proofs use the assumption that $\lambda_i = 1/n$ , for all integers $1 \le i \le n$ .

Proof by Rearrangement

Define the $n$ sequence $\{ r_{ij}\}_{i=1}^{n}$ as $r_{ij} = \sqrt[n]{a_i}$ , for all integers $1 \le i,j \le n$ . Evidently these sequences are similarly sorted. Then by the Rearrangement Inequality, $\sum_i a_i = \sum_i \prod_j r_{ij} \ge \sum_i \prod_j r(i,i+j) = n \prod_i \sqrt[n]{a_i} ,$ where we take our indices modulo $n$ , with equality exactly when all the $r_{ij}$ , and therefore all the $a_i$ , are equal. Dividing both sides by $n$ gives the desired inequality. $\blacksquare$

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Proofs of AM-GM

Contents

Complete Proofs

Proof by Convexity

Proofs of Unweighted AM-GM

Proof by Rearrangement