Folland Theorem 6.20 (Application Minkowski Inequality Integrals)

Question

I am attempting to understand Folland Theorem 6.20, which uses the Minkowski Inequality for Integrals. Both are attached below. I understand up until "Therefore, by Minkowski's Inequality for Integrals..." However, I do not understand how the given function fulfills the premises of Minkowski or how it is being applied in the line below.

Thanks!