A characterization of a nonlinear integral triangle inequality

Let $(E,\|.\|)$ be a Banach space and let $(\Omega,\mu)$ be a Lebesgue measure space. We characterize, for all $p>0$, measurable functions $u:\Omega\rightarrow \mathbb{R}$ for which \begin{equation*} \left\| \int_{\Omega} f\,d\mu \right\|^{p}\,\leq\,\int_{\Omega} u \| f \|^{p}\,d\mu.\tag{I} \end{equation*} We characterize $u$ for the reverse of (I) as well. The discrete counterpart of this problem is also solved.