1. ## Use Hölder's inequality

Let $(X, \mathcal{B}, \mu)$ be a measure space. Use Hölder's inequality to prove that if $\frac{1}{p}+\frac{1}{q} = \frac{1}{r}$ and $r \geq 1$ then for every $f \in L^p(X, \mathcal{B}, \mu)$, $g \in L^q(X, \mathcal{B}, \mu)$, and $fg \in L^r(X, \mathcal{B}, \mu)$ with $|| fg ||_r \leq || f ||_p \cdot || g ||_q$.

2. Originally Posted by Erdos32212
Let $(X, \mathcal{B}, \mu)$ be a measure space. Use Hölder's inequality to prove that if $\frac{1}{p}+\frac{1}{q} = \frac{1}{r}$ and $r \geq 1$ then for every $f \in L^p(X, \mathcal{B}, \mu)$, $g \in L^q(X, \mathcal{B}, \mu)$, and $fg \in L^r(X, \mathcal{B}, \mu)$ with $|| fg ||_r \leq || f ||_p \cdot || g ||_q$.

I don't see how to prove this right now. I know that this looks a lot like Young's inequality which follows from Hölder's inequality, so I was thinking to adopt a similar proof of that theorem. Right now, I just need a few hints on how to proceed. Thank you.
Simply apply Hölder's inequality with the correct p and q... Write down what you want to prove ( $(\int (f g)^r)^{1/r}\leq \cdots$), rewrite it like Holder's inequality ( $\int (f^r) (g^r) \leq \cdots$) in order to guess the exponents, and its shouldn't be a problem.