= Solution
If either upper norm bound is nonpositive, there are no admissible functions with $\|f\|_q\geq C_q>0$, and the assertion is vacuous. For positive $C_p,C_r$, choose
$$
\varepsilon=\left(\frac{C_q^q}{2C_p^p}\right)^{1/(q-p)},\qquad A=\{x:|f(x)|>\varepsilon\}.
$$
Splitting the integral gives
$$
C_q^q\leq\|f\|_q^q
\leq\varepsilon^{q-p}\|f\|_p^p+\int_A|f|^q
\leq\frac{C_q^q}{2}+\int_A|f|^q.
$$
For finite $r$, <Holder inequality> bounds the last term by $\|f\|_r^q\mu(A)^{1-q/r}\leq C_r^q\mu(A)^{1-q/r}$. Hence
$$
\mu(A)\geq\left(\frac{C_q^q}{2C_r^q}\right)^{r/(r-q)}.
$$
For $r=\infty$, the corresponding bound is $\int_A|f|^q\leq C_r^q\mu(A)$. Thus an explicit choice proving the strict inequality is
$$
\boxed{M=
\begin{cases}
\frac12\left(C_q^q/(2C_r^q)\right)^{r/(r-q)},&r<\infty,\\
C_q^q/(4C_r^q),&r=\infty.
\end{cases}
\quad\mu\{|f|>\varepsilon\}>M.}
$$
This is a <nonvanishing level-set bound between three Lp norms>: the lower-$p$ bound prevents diffuse low-amplitude mass, and the higher-$r$ bound prevents arbitrarily concentrated spikes.
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