I wonder about the Maclaurin series of the analytic $f(z)$ such that
$$ f(f(z)) = g(z) = \exp(z) - \exp(-z) + \exp(-z/2) - \exp(-z/5)+ \exp(-z/6) - \exp(-z/9) $$
Since zero is a fixpoint and $g'(0) = \frac{74}{45} > 1$ we can find such a Maclaurin series of $f(z)$ in many ways.
Using Koenigs function for instance. Or by using Carleman matrices.
We then can compute
$$f(z) = a_1 z + a_2 z^2 + a_3 z^3 + ...$$
I would like many terms of these coefficients and I wonder how many terms we need before one of them is negative ?
I also wonder about the exact radius of $f(z)$. Ofcourse the radius is equal or less than the distance to the next fixpoint of $g(z)$.
I know there is a fixpoint for $g$ around $-0.218 + 1.504 i$ and its complex conjugate.
Is the radius equal to the closest distance to a point $A$ where $f(f(A))=g(A)$ still holds ?