How to show that $(\int\left(\sum_k|f_k|\right)^p)^{1/p}\le\int(\sum_k|f_k|^p)^{1/p}$, or is it not true in general ?
$\{f_k\}\subset L^p, G_n=\sum_1^n|f_k|$, I think $|G_n|_p$ is the sum above on the left, and $\sum|f_k|_p$ is on the right, is it sufficient to argue that; since $|.|_p$ is a norm the inequality must hold.
I would rather say that the opposite is true, and that it comes from Jensen's inequality: since $x \mapsto x^{1/p}$ is concave for $p>1$, one has: $$ \left( \int g \right)^{1/p} \geq \int \left(g^{1/p}\right).$$ Then you apply this result to the function $g = \sum_k \vert f \vert^p$.