For any real $\alpha > 0$, consider $J _ { \alpha } = 2 \ln ( 2 ) - \ln \left( 1 + \alpha ^ { 2 } \right) - 2 \alpha \arctan \left( \frac { 1 } { \alpha } \right)$. Show that there exists $\gamma > 0$ such that, for all $\alpha \in ] 0 , \gamma [$, $J _ { \alpha } > 0$.