For what measures $\mu$ and what intervals $(a,b) \subset (1,\infty)$ is the function
$p\mapsto \|f\|_p =\left(\int |f|^p d\mu \right)^{2/p}$
a convex function of $p$ on $(a,b)$ for all $f\in L^1(\mu) \cap L^\infty(\mu)$? When is it concave?
For what measures $\mu$ and what intervals $(a,b) \subset (1,\infty)$ is the function
$p\mapsto \|f\|_p =\left(\int |f|^p d\mu \right)^{2/p}$
a convex function of $p$ on $(a,b)$ for all $f\in L^1(\mu) \cap L^\infty(\mu)$? When is it concave?