I encounter a question as follow:
Given $p \leq q$ and $f$ is $C^n$ function, find a condition (if any) on $\mu$ ($\mu$ is finite measure) such that the following inequality holds:
$[\int \left(|f(x)|^{p} + \sum_{i=1}^n |f^{(i)}(x)|^{p}\right)\mu(dx)]^{1/p}\leq [\int \left(|f(x)|^{q} + \sum_{i=1}^n |f^{(i)}(x)|^{q}\right) \mu(dx)]^{1/q}$
Could someone provide an answer for the above question? From my point of views, I guess it is not true in general and I would like to seek others idea.