Let $\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4}$ be four strictly positive real numbers pairwise distinct and two strictly positive real numbers $E$ and $N$. Let $\Omega$ be the part, assumed to be non-empty, formed of the quadruplets $x = (x_{1}, x_{2}, x_{3}, x_{4})$ of $\mathbb{R}_{+}^{4}$ satisfying: $$\left\{\begin{array}{l} x_{1} + x_{2} + x_{3} + x_{4} = N \\ \varepsilon_{1} x_{1} + \varepsilon_{2} x_{2} + \varepsilon_{3} x_{3} + \varepsilon_{4} x_{4} = E \end{array}\right.$$ We define the function $F$ for all $x = (x_{1}, x_{2}, x_{3}, x_{4})$ of $\mathbb{R}_{+}^{4}$ by $$F(x_{1}, x_{2}, x_{3}, x_{4}) = -\sum_{i=1}^{4} \ln \Gamma(1 + x_{i})$$ We suppose that there exists $\bar{N} = (N_{1}, N_{2}, N_{3}, N_{4}) \in \Omega$, the numbers $N_{1}, N_{2}, N_{3}, N_{4}$ all being non-zero, such that $$\max_{x \in \Omega} F(x) = F(\bar{N})$$ Show the existence of two real numbers $\lambda$ and $\mu$ satisfying for all $i \in \{1,2,3,4\}$: $$\ln N_{i} + \frac{1}{2N_{i}} + \int_{0}^{+\infty} \frac{h(u)}{(u+N_{i})^{2}} du = \lambda + \mu \varepsilon_{i}$$