Can a function $f$ have $L^p$ norms $\Vert f\Vert_p =p$ for all $1\leq p<\infty$? I have tried to show that such a function must be in $L^{\infty}$ and thus it is impossible for such a function to exist since, in that case, $$\infty =\lim_{p\rightarrow \infty} \Vert f\Vert_p=\Vert f\Vert_{\infty}.$$
Basic estimates seem to fail and I can't seem to construct a counterexample.
What if we replace $p$ with something that grows at different rates, like $e^p$ or $\log p$?

The magic words are: Riesz-Thorin interpolation Theorem.
