Young's Inequality for Products/Proof by Convexity

Proof
The result follows directly if $a = 0$ or $b = 0$.

, assume that $a > 0$ and $b > 0$.

Then:

By, the equality occurs :
 * $\map \ln {a^p} = \map \ln {b^q}$

That is, :
 * $b = a^{p-1}$