Minkowski's Inequality for Integrals/Equality

Theorem
Let $f, g$ be (Darboux) integrable functions.

Let $p \in \R$ such that $p > 1$.

Then equality in Minkowski's Inequality for Integrals, that is:
 * $\ds \paren {\int_a^b \size {\map f x + \map g x}^p \rd x}^{1/p} = \paren {\int_a^b \size {\map f x}^p \rd x}^{1 / p} + \paren {\int_a^b \size {\map g x}^p \rd x}^{1 / p}$

holds, for all $x \in \closedint a b$:
 * $\dfrac {\map f x} {\map g x} = c$

for some $c \in \R_{>0}$.