Suppose I have a vector space upon which I have defined the $l_p$ norm. If I define a function
$$f(x) = \frac{||x||_p^{n+1}}{||x||_q^{n}}$$
then what are the conditions on on $p$, $q$ and $n$ for $f$ to be a norm? Trivially $n=1$ works, but I'm not sure how to prove for higher $n$.
Specifically, I need to prove that $f(x+y) \leq f(x) + f(y)$. Alternatively, if this is a known norm which has a name then I can go look it up.