Weighted arithmetic-geometric mean inequality (${\prod }_i{a}_i^{w_i}\le {\sum }_i{w}_i{a}_i$)

For nonnegative real numbers $a_i$ and positive weights $w_i$ with ${\sum }_i{w}_i=1$, the weighted geometric mean does not exceed the weighted arithmetic mean: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 $a^s{b}^{1-s}\le sa+(1-s)b$ when $0<s<1$.