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