In Theorem 2.6.4 of Beardon's book, "Iteration of Rational Functions", he states the values for the first two coefficients of an iterated power series.
That is, suppose that $$ f(z)=az+b_{1}z^{r+1}+\cdots $$ near the origin, where $a, b_{1} \neq 0$ and $r \geq 1$. Then $$ f^{n}(z)=a^{n}z+b_{n}z^{r+1}+\cdots, $$ where $$ b_{n} = a^{n-1}b_{1} \left( 1+a^{r}+a^{2r}+\cdots+a^{(n-1)r} \right). $$
My questions are the following.
(1) are there any known explicit such results?
I.e., if $|z| \leq c_{1}(f,n)$, then
$$
\left| f^{n}(z) - \left( a^{n}z+b_{n}z^{r+1} \right) \right| <c_{2}(f,n)|z|^{r+2},
$$
where $c_{1}(f,n)$ and $c_{2}(f,n)$ are explicit.
I imagine a general such result is far too much to ask for, so perhaps a result for $f(z)$ a rational function whose numerator and denominator are of small degree (both of degree at most $4$ would actually work for my needs).
(2) are there known techniques for handling such problems that can be applied to specific cases?
Any references, ideas or other suggestions would be greatly appreciated.
