Let $\Omega$ be a limited domain of $\mathbb{R}^n$. Suppose $u$ to be measurable over $\Omega$, and suppose that for every $p \geq 1$
$$|u|^p \in L^1(\Omega)$$
Now, defined
$$\phi_p(u) = \left(\frac{1}{\text{measure}(\Omega)}\int_{\Omega}|u|^p\ \text{d}x\right)^{1/p}$$
prove that for every $(p, q) \geq 1$ with $p \leq q$ we have
$$\Phi_p(u) \leq \Phi_q(u)$$
Any hint? Thank you so much!
Hint:
If $p = q$, there is nothing to prove, so suppose $p < q$. Apply Hölder's inequality with conjugate exponents $q/p$ and $q/(q-p)$.