I'm stuck on part of a real analysis problem; unfortunately, I'm stuck with how to start it.
Suppose $f \in L^p(X)$ for all $p$ satisfying $r < p < s$. Let $\phi(p) = ||f||_p^p$. Show that $\phi$ is log-convex on $(r,s)$.
Other parts of the problem ask for continuity of $\phi$, the connectedness (i.e. convexity) of the set of values $p$ at which $\phi$ is finite, that $\phi(p) \rightarrow ||f||_\infty$ and the inclusion $L_r \cap L_s \subset L_p$, all of which I have done, but I am just totally stuck on this other part. Indeed, I can't even show that $\phi$ is convex, much less log-convex.
This problem is presented right after Holder's Inequality, so it should not use any advanced machinery or differentiability. I could really use some help, here!