1. ## Lp Space Question

Let $(X, \mathcal{A}, \mu)$ be a measure space. Prove that if $f$ belongs to $L^4(\mu)$ and $L^8(\mu)$ then it belongs to $L^6(\mu)$ with $||f||_6 \leq ||f||_4^{\frac{1}{3}} \cdot ||f||_8^{\frac{2}{3}}$.

$L^p(\mu)$ is contains equivalence classes where $\int |f|^p d\mu < \infty$.
Also, $||f||_p=(\int |f|^p d\mu)^{\frac{1}{p}}$.

Attempt
Let $f \in L^4(\mu)$ and $L^8(\mu)$. Then

$\int |f|^4 d\mu < \infty$ and $\int |f|^8 d\mu < \infty$

I don't see how to reach the conclusions now. I would appreciate any help on how to proceed. Thanks in advance.

2. $||f||_6^6=||f^4 f^2||_1\leq ?$

Think of the French guy with the German for black...

3. CBS for the win