= Weighted arithmetic-geometric mean inequality
{title2=$\prod_i a_i^{w_i}\leq\sum_i w_i a_i$}
For nonnegative real numbers $a_i$ and positive weights $w_i$ with $\sum_iw_i=1$, the weighted <geometric mean> does not exceed the weighted <arithmetic mean>:
$$
\prod_i a_i^{w_i}\leq\sum_iw_i a_i.
$$
For positive $a_i$, this follows from <Jensen's inequality> and the <concavity> of the <natural logarithm>; a <limit> extends it to zero inputs. For two terms it gives $a^sb^{1-s}\leq sa+(1-s)b$ when $0<s<1$.
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