$p$ and $q$ are positive interger and $\frac{1}{p}+\frac{1}{q}=1$
For $0 \le u$ and $0\le v$
prove that $uv \le \frac{u^p}{p}+\frac{v^q}{q}$.
Put $f(x)=x^{p-1}, f^{-1}=y^{q-1}$ then
$uv \le \int_{0}^{u^p} f(x)dx+ \int_{0}^{v^q} f^{-1}(y)dy$
Why this inequality holds?