Reverse Young's Inequality for Products

Theorem
Let $p, q \in \R_{> 0}$ be strictly positive real numbers satisfying:
 * $\displaystyle \frac 1 p - \frac 1 q = 1$

Let $a \in \R_{\ge 0}$ be a positive real number.

Let $b \in \R_{> 0}$ be a strictly positive real number.

Then:
 * $\displaystyle a b \ge \frac {a^p} p - \frac {b^{-q}} q$

Proof
Define:

Then:

Thus Young's Inequality for Products can be applied: