Young's inequality (for products) and the AM-GM inequality: random musings
Young's inequality (for products) is sometimes called the generalised AM-GM (arithmetic mean -geometric mean) inequality . Let $a$ and $b$ be non-negative real numbers. Then $(ab)^{1/2} \leq (a+b)/2$, with equality if and only if $a=b$. This can be seen easily by multiplying both sides by $2$, then squaring both sides and noting that $(a+b)^2=(a-b)^2+4ab$. This is the special case of the AM-GM inequality for two non-negative real numbers (and the result generalises to any finite number of non-negative real numbers). However it is also the special case of Young's inequality when $p=q=2$. Young's inequality states that for non-negative real numbers $x$ and $y$ and positive real numbers $p$ and $q$ satisfying $1/p + 1/q = 1$, we have $xy \leq x^p/p + y^q/q$, with equality if and only if $x^p=y^q$. When $p=q=2$ this says that $xy \leq (x^2+y^2)/2$ with equality if and only if $x^2=y^2$, which is the same as the version of AM-GM above above when you set $a=x^2$ and $b=y^2$...