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!
Thanks!
Fix $t_1,t_2\in(r,s)$ and $\lambda\in(0,1)$. Then we have $$\log\phi(\lambda t_1+(1-\lambda)t_2)=\log\left(\int|f|^{\lambda t_1+(1-\lambda)t_2}\right) $$ By Holder's inequality, we have \begin{align*} \log\left(\int|f|^{\lambda t_1+(1-\lambda)t_2}\right)& \leq\log\left[ \left( \int |f|^{t_1}\right)^\lambda \left( \int |f|^{t_2}\right)^{1-\lambda}\right]\\ &=\lambda\log\left( \int |f|^{t_1} \right)+(1-\lambda)\log\left( \int |f|^{t_2} \right)\\ &=\lambda\log\phi(t_1)+(1-\lambda)\log\phi(t_2) \end{align*} and therefore $\log\phi$ is convex.