Let $f(x)=\sum_{n\geq 1} c_n\cdot x^n$ be a function given by a power series. Further there is some $\alpha >1$ such that for all $n$, $c_n = \Theta(1/n^{\alpha})$. What can one say about the asymptotics (in terms of n) of the coefficients of the Taylor expansion (around $0$) of the inverse of f assuming it exists? For instance, is the $n$-th coefficient less than $1/\beta^n$ for some $\beta>0$?

I couldn't get much mileage out of trying to analyze the Lagrange-Burrman formula or Bell polynomials.