Integral mean inequalities

Question

If $f \in C[0,1]$, then should be true that $$\left( \int |f|^p\right)^{1/p} \leq \left( \int |f|^q\right)^{1/q}$$ for $1<p \leq q$. However, I have found no sources on this fact. Is it true?

Answer 1

This follows from Hölder's inequality. Letting $r$ denote the Hölder conjugate of $\frac{q}{p}$, we have $$\|f\|_p^p = \|f^p\|_1 \leq \|f^p\|_{q/p}\|1\|_r = \|f^p\|_{q/p} = \|f\|_q^p$$ so $\|f\|_p \leq \|f\|_q$. More generally, if $(X, \mathcal{M}, \mu)$ is a finite measure space, the same proof shows that $\|f\|_p \leq \|f\|_q\mu(X)^{\frac{1}{p}-\frac{1}{q}}$.

Answer 2

This is also a direct consequence of Jensen's inequality

Since $\varphi(x)=x^\frac{q}{p}$ is convex, we obtain

$$ \left(\int |f|^p\right)^{\frac{q}{p}}\varphi\left(\int |f|^p\right) \leq \int \varphi(|f|^p) = \int |f|^q$$ Taking the $q$-th root shows the result.