For all $i \in \{1,2,3,4\}$, we set
$$\theta(N_{i}) = \frac{1}{2N_{i}} + \int_{0}^{+\infty} \frac{h(u)}{(u+N_{i})^{2}} du$$
\textbf{VI.C.1)} Show that for all $i \in \{1,2,3,4\}$
$$0 < \theta(N_{i}) < \frac{1}{N_{i}}$$
\textbf{VI.C.2)} Show the existence of a strictly positive real $K$ such that for all $i \in \{1,2,3,4\}$
$$N_{i} = K e^{\mu \varepsilon_{i}} e^{-\theta(N_{i})}$$