Let \((X, \mathscr{A}, \mu)\) be a measure space and \(p \in(0,1)\). The conjugate index is given by \(q:=\) \(p /(p-1)<0\). Prove for all measurable \(u, v, w: X ightarrow(0, \infty)\) with \(\int u^{p} d \mu, \int v^{p} d \mu<\infty\) and \(0<\int w^{q} d \mu<\infty\) the inequalities

\[\int u w d \mu \geqslant\left(\int u^{p} d \muight)^{1 / p}\left(\int w^{q} d \muight)^{1 / q}\]

and

\[\left(\int(u+v)^{p} d \muight)^{1 / p} \geqslant\left(\int u^{p} d \muight)^{1 / p}+\left(\int v^{p} d \muight)^{1 / p} .\]

[ consider Hölder's inequality for \(u\) and \(1 / w\).]