Log-convexity of the p-norms of a fixed function
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.