A characterization of a nonlinear integral triangle inequality

Let $(E,\mathrm{\parallel }\mathrm{.}\mathrm{\parallel })$ be a Banach space and let $(\mathrm{\Omega },\mu )$ be a Lebesgue measure space. We characterize, for all $p>0$, measurable functions $u:\mathrm{\Omega }\to \mathbb{R}$ 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.