Result 0.0.1  The $f_m^{\eta }(\mathbb{z})$ can be expanded as a geometric series for $\mathbb{z}<<1$
$$f_m^{\eta }(\mathbb{z})\approx \sum _{\alpha =1}^{\mathrm{\infty }}{\eta }^{\alpha +1}\frac{{\mathbb{z}}^{\alpha }}{{\alpha }^m}=\mathbb{z}+\eta \frac{{\mathbb{z}}^2}{2^m}+\frac{{\mathbb{z}}^3}{3^m}+\cdots$$