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. 



Asymptotic growth of the of Taylor coefficients of the inverse of a function
You can use the Faa-di-Bruno formula, like it is done on page 7 of [here](http://www.mat.univie.ac.at/~michor/diff-zoo.pdf)
or the proof of claim (b) on page 21 of [here](http://www.mat.univie.ac.at/~michor/exotic-zoo.pdf).


