$\int|f|^\alpha|g|^{1-\alpha}d\mu\le(\int|f|d\mu)^\alpha(\int|g|d\mu)^{1-\alpha}$

Question

$f,g$ on $(\Omega, \mathcal{A}, \mu)$ and $0<\alpha<1$.

Then $$ \int|f|^\alpha |g|^{1-\alpha} d\mu \le \left(\int|f|d\mu\right)^\alpha \left(\int|g|d\mu\right)^{1-\alpha} $$

This seems like some case of Holder's inequality but I haven't found anything about it, is this inequality correct? Does anyone have a link to a proof of this? I would appreciate it! Thanks!

Answer 1

The numbers $p = \dfrac 1\alpha$ and $q = \dfrac 1{1-\alpha}$ are Holder conjugates, so that $$\int |f|^\alpha |g|^{1-\alpha} \le \left( \int |f|^{\alpha \frac 1\alpha} \right)^\alpha \left(\int |g|^{(1-\alpha) \frac{1}{1-\alpha}} \right)^{1 - \alpha} = \left( \int |f| \right)^\alpha \left( \int |g| \right)^{1-\alpha}.$$