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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.

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    $\begingroup$ To talk about inverse function at $0$ you need first of all that $f(0)=0$, that is $c_0=0$. Then the rate of coefficients of the inverse function is regulated by the radius of the largest disk centered at $0$ in which this inverse function exists. And this has nothing to do with the rate of $c_n$. Even when $c_n$ is a finite sequence, the rate of the coefficients of the inverse can be anything. $\endgroup$
    – Alexandre Eremenko
    Nov 1, 2017 at 13:24
  • $\begingroup$ Great - thanks for the information. I had a more specific function in mind, and the radius of the disk you mentioned gave me the bound I desire. $\endgroup$
    – user116726
    Nov 1, 2017 at 18:18

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You can use the Faa-di-Bruno formula, like it is done on page 7 of here or the proof of claim (b) on page 21 of here.

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  • $\begingroup$ Thanks - I think I'm able to understand the gist of what's going on. The reference is a little notation heavy given my background, so I just wanted to confirm the following: What is the dependence of beta on alpha? (any bound at all is ok, I don't need to know the optimal dependence). $\endgroup$
    – user116726
    Nov 1, 2017 at 10:07

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