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

$\displaystyle L^p(\mu)$ is contains equivalence classes where $\displaystyle \int |f|^p d\mu < \infty$.

Also, $\displaystyle ||f||_p=(\int |f|^p d\mu)^{\frac{1}{p}}$.

Attempt

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

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

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