In the study of a particular PDE I found myself wanting to prove the following inequality:

$( \int_0^{\infty} r^{-3} |f|^6 \; dr )^{1/6} \leq C ( \int_0^{\infty} [ r^{-1} |f|^2 + r |f'|^2 + r |f''|^2] \; dr )^{1/2}$

for some constant $C > 0$ and all $f \in C^{\infty}_c((0,\infty);(-\infty,\infty))$.

Partial progress:

If $\mathop{supp} f \in [r_0, \infty)$, for some $r_0 > 0$ then $( \int_0^{\infty} r^{-3} |f|^6 \; dr )^{1/6} \leq C(r_0) ( \int_0^{\infty} [ r^{-1} |f|^2 + r |f'|^2 ] \; dr )^{1/2}$. However, $C(r_0) \rightarrow \infty$ as $r_0 \searrow 0$ as can be seen by basic scaling. This lead me to try to decompose $f$ into two parts, for concreteness, one supported on say $[1/2,\infty)$ and the other supported in $(0,2]$. However, this didn't lead me anywhere.

I employed a heuristic analysis as in http://terrytao.wordpress.com/2010/03/11/a-type-diagram-for-function-spaces/ . Of course, one has to also track the position of the bump function since the norms are not translation invariant. The inequality was satisfied under this heuristic but I didn't know how to make it rigourous.

Tried the change of variables $p = \log r$ to no avail.

I don't have any reason, except for the above, to believe that the inequality is true so perhaps someone can come up with a counterexample?

Any help would be much appreciated, thanks in advance.