Minkowski’s Integral Inequality for Function Norms

Let ρ and λ be Banach function norms with the Fatou property. Then the generalized Minkowski integral inequality ρ(λ(f x )) ≤ Mλ(ρ(f y )) holds for all measurable functions f(x,y) and some fixed constant M if and only if there exists 1 ≤ ρ ≤ ∞ such that λ is p-concave and ρ is p-convex.