We consider a power series $g \in O_1$ with non-negative real coefficients. We assume that there exist $a > 0, b > 0$ such that
$$g \prec a\left(I + \frac{g^2}{b - g}\right)$$
Show that there exist $r > 0$ and a function $h: ]-r, r[ \longrightarrow \mathbb{R}$, expandable as a power series at 0, satisfying $h(0) = 0$ and such that
$$h(x) = a\left(x + \frac{h(x)^2}{b - h(x)}\right)$$
for all $x \in ]-r, r[$. We also denote by $h$ the element of $O_1$ associated with the function $h$.