Show that there exists a real $\alpha > 0$ such that
$$\forall \theta \in [ - \pi , \pi ] , 1 - \cos \theta \geq \alpha \theta ^ { 2 }$$
Deduce that there exist three reals $t _ { 0 } > 0 , \beta > 0$ and $\gamma > 0$ such that, for all $t \in ] 0 , t _ { 0 } ]$ and all $\theta \in [ - \pi , \pi ]$,
$$\left| \frac { P \left( e ^ { - t } e ^ { i \theta } \right) } { P \left( e ^ { - t } \right) } \right| \leq e ^ { - \beta \left( t ^ { - 3 / 2 } \theta \right) ^ { 2 } } \quad \text { or } \quad \left| \frac { P \left( e ^ { - t } e ^ { i \theta } \right) } { P \left( e ^ { - t } \right) } \right| \leq e ^ { - \gamma \left( t ^ { - 3 / 2 } | \theta | \right) ^ { 2 / 3 } }$$