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Let $ f(x) = \sum a_n x^n$ Such that The $a_n$ are real and $f(a),f ' (a) , f " (a) > 0 $ for any real $a > 0$.

Let $ \ln(f(exp(x))) = \sum b_n x^n $.

Let $c_n = a_n - b_n$.

For a given $f$ and a given $y$ When is it true that $c_n ^2 < (y / n )^2$ (for every n) and how to prove this ?

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  • $\begingroup$ I know faa di Bruno , but not sure how that helps. $\endgroup$
    – mick
    Feb 25, 2016 at 15:33
  • $\begingroup$ You postulate that series $\sum b_n x^n$ exists? I do not see how it follows from the other assumptions. $\endgroup$
    – GEdgar
    Feb 25, 2016 at 15:56
  • $\begingroup$ @GEdgar : notice that $f(0) > 0 , exp(0)=1 $and $f(1) > f(0)$. Therefore $\ln(f(exp(0))) $ exists and thus we have a Taylor series. At least a formal one. The radius is another matter, but that relates to the coëfficiënts ofcourse. $\endgroup$
    – mick
    Feb 25, 2016 at 19:23
  • $\begingroup$ I also know Carleman matrices , but again I do not know how and if that helps. $\endgroup$
    – mick
    Feb 25, 2016 at 19:31
  • $\begingroup$ Maybe if we consider $f$ as a polynomial first ? $\endgroup$
    – mick
    Feb 26, 2016 at 12:22

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