How can I prove Holder inequality for $0<p<1$? [closed]
Write $h(x) = f(x)g(x)$. Without loss of generality (by multiplying $g$ and $h$ by constants) we can assume $$ \left(\int |g|^{p'} dx \right)^{\frac{1}{p'}}= 1 = \int |h| dx $$ So what we want to prove reduces to $$ \int |h / g|^p \leq 1 $$
To prove this, apply the regular Holder inequality: put $h^p$ in $L^{1/p}$ and $1/g^p$ in $L^{1/1-p}$. Observe that
$$ \frac{1}{|g|^{p/(1-p)}} = |g|^{p'}$$