Pólya’s Proof of the Weighted Arithmetic–Geometric Mean Inequality

This article provides a formalisation of the Weighted Arithmetic–Geometric Mean Inequality: given non-negative reals $a_1, \ldots, a_n$ and non-negative weights $w_1, \ldots, w_n$ such that $w_1 + \ldots + w_n = 1$, we have \[\prod\limits_{i=1}^n a_i^{w_i} \leq \sum\limits_{i=1}^n w_i a_i\ .\] If the weights are additionally all non-zero, equality holds if and only if $a_1 = \ldots = a_n$.

As a corollary with $w_1 = \ldots = w_n = 1/n$, the regular arithmetic–geometric mean inequality follows, namely that \[\sqrt[n]{a_1\,\cdots\, a_n} \leq \tfrac{1}{n}(a_1 + \ldots + a_n)\ .\]

I follow Pólya's elegant proof, which uses the inequality $1 + x \leq e^x$ as a starting point. Pólya claims that this proof came to him in a dream, and that it was “the best mathematics he had ever dreamt.”