Let us define the $L^p$ norm of a function $f:\mathbb{R} \to \mathbb{R}$ as $$ \|f\|_p:= \left(\int \lvert f \rvert^p\right)^{1/p}. $$
Let us label the following statement by $S\left(p\right)$: $$ \|f\|_p \leq \|g\|_p. $$
My Question: Suppose $p>q>0$. Then, does there exist any interrelationship between $S\left(p\right)$ and $S\left(q\right)$? To be precise, does one implies the other?
I think neither implies the other, yet I cannot find a concrete counterexample (I was working with $p=2$ and $q=1$). Also, I was wondering if such implication holds if we assume stronger assumptions on $f$ and $g$. Any help will be much appreciated. Thank you.
Let $f=I_{[0,1]}$, i.e., the indicator function on $[0,1]$. Similarly, let $g=\frac{1}{\lambda}I_{[0,\lambda]}$. Then $f$ and $g$ have the same $L^1$ norm, but their $L^2$ norms are $1$ and $\sqrt{\frac{\lambda}{\lambda^2}}$. Choosing values for $\lambda$ that are greater than or less than $1$ gives norms both greater than or less than $1$ for $g$.
Therefore, you need to severely restrict your function $f$ and $g$ to have any hope of a relationship between their norms. There are some things you can say if, say, you knew that $f$ had a smaller $L^p$ and $L^q$ norm than $g$ and wanted an inequality on $L^r$ where $r$ is between $p$ and $q$.
Additional information can be found on the Wikipedia page on $L^p$ spaces.
In addition, Hölder's inequality can be used to prove log-convexity of the $L^p$ norm, which is related, but not equivalent, to your question.