I want to find out for which $p, q$ exists $C > 0$ with $||f||_p \leq C ||f||_q$ for all $f \in C([0,1]), p \in \mathbb{R}_{\geq 1} \cup \{\infty\}$. I first let $p,q \geq 1$ and I looked at the case where $q > p$.
I defined $\tilde{f} := |f|^p$ and $\tilde{p} := \frac{q}{p}$, $f \in C([0,1])$ of course. Also, I let $g(x) = 1$ and applied Hölder's inequality:
$||\tilde{f} \cdot g||_1 \leq ||\tilde{f}||_\tilde{p} \cdot ||g||_\tilde{q}$
where of course $||g||_\tilde{q} = 1$.
$\implies \int_0^1 |f|^p \mathrm dx \leq \left ( \int_0^1 |f|^q \mathrm dx \right )^\frac{p}{q} \iff ||f||_p \leq ||f||_q \cdot C \;\; \forall C \geq 1$
So my solution is, that for $q > p \geq 1$, there always exists $C > 0$ such that the inequality holds good. Now, before I look at $p > q$, I wanted to know whether my solution is correct. It doesn't seem to be complete to me, as there might also exist $C \in (0,1)$ such that the inequality holds good. Am I wrong? If not, how can I find out for which $p$ and $q$ there exists a $C \in (0,1)$?
Thanks in advance for any answers...