\begin{equation}
    \frac{\sigma^2}{\hat{q}^2} + \frac{1}{\hat{q}^2} \int_{-1}^{\hat{q}} \widetilde{q}^2 \nu(\mathrm{d}\widetilde{q}) + \int_{\hat{q}^{+}}^{\infty} \nu(\mathrm{d}\widetilde{q}) \geq \frac{\left(\frac{\sigma^2}{\hat{q}} + \frac{1}{\hat{q}} \int_{-1}^{\hat{q}} \widetilde{q}^2 \nu(\mathrm{d}\widetilde{q}) + \int_{\hat{q}^{+}}^{\infty} \widetilde{q} \nu(\mathrm{d}\widetilde{q}) \right)^2}{\sigma^2 +  \int_{-1}^{\hat{q}} \widetilde{q}^2 \nu(\mathrm{d}\widetilde{q}) + \int_{\hat{q}^{+}}^{\infty} \widetilde{q}^2 \nu(\mathrm{d}\widetilde{q})}. \notag
\end{equation}
