Young's Inequality for Products

Theorem
Let $p, q \in \R_{> 0}$ be strictly positive real numbers such that:
 * $\dfrac 1 p + \dfrac 1 q = 1$

Then, for any $a, b \in \R_{\ge 0}$:
 * $a b \le \dfrac {a^p} p + \dfrac {b^q} q$

Equality occurs :
 * $b = a^{p - 1}$