Is $\frac{||f||_{L^p}^p}{||f||_{L^1}^2} \leq K ||f||_{L^r}^{s}$ possible?

Let $\Omega=(0,1) \subset \mathbb{R}$ (or, $\Omega$ is any open set in $\mathbb{R}^d$).

Let's consider the usual $L^p(\Omega)$ space with the Lebesgue measure. I'm only assuming $f \geq0$ and $f \in L^p(\Omega)$, where $p>3$.

My question is:Can we obtain an estimate like$\frac{||f||_{L^p}^p}{||f||_{L^1}^2} \leq K ||f||_{L^r}^{s}$?Here, $K$ is constant and $r \leq p$.

For example, if $f \equiv c$, then $r,s$ can be $p-2$.

If there exists a counterexample, please let me know. Thank you.