Definition:
Let $a,b\in \mathbb{R}$ with $a<b$ and for $f\in C([a,b])$, $1\leq p <\infty$ let
$$||f||_p=\left(\int_{a}^{b}|f(x)|^p \, dx\right)^{1/p} \text{and } | |f||_\infty := \max\limits_{x\in [a,b]}|f(x)|$$
To Show: $||f||_p\leq (b-a)^{c_{p,q}}||f||_q$ with $c_{p,q}=\begin{cases}\frac{q-p}{q} \quad q<\infty ; \\ \frac{1}{p} \quad \quad q=\infty\end{cases}$
I think I already showed $q=\infty$ by using $\int_{a}^{b}|f(x)|^p \, dx\leq (b-a)||f||_\infty^p$ which implies that $||f||_p=\left(\int_{a}^{b}|f(x)|^p \, dx\right)^{1/p}\leq (b-a)^{1/p}||f||_\infty$.
But how do I show $q<\infty$?