Define the following function from $(0,1] \times \mathbb{R}_+^\star \times \mathbb{R}$ to $\mathbb{R}_+ \times \mathbb{R}^2$ :
$$ G : (p, \lambda, b) \longmapsto \begin{pmatrix} p \lambda \mathbb{E} \left[ g'(\lambda Z + b) \right] \\ p \lambda^2 \mathbb{E} \left[ g''(\lambda Z + b) \right] \\ p \lambda^3 \mathbb{E} \left[ g^{(3)}(\lambda Z + b) \right] \end{pmatrix},$$where $g (x) = \frac{e^x}{1+e^x}$ and $Z \sim \mathcal{N}(0,1)$.
Any clue to show that $G$ is injective ?
I have tried global inversion, but it turns out that $G$ is not proper.