# Taylor of $\ln(f(exp(x))))$?

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 ?

• I know faa di Bruno , but not sure how that helps. – mick Feb 25 '16 at 15:33
• You postulate that series $\sum b_n x^n$ exists? I do not see how it follows from the other assumptions. – GEdgar Feb 25 '16 at 15:56
• @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. – mick Feb 25 '16 at 19:23
• I also know Carleman matrices , but again I do not know how and if that helps. – mick Feb 25 '16 at 19:31
• Maybe if we consider $f$ as a polynomial first ? – mick Feb 26 '16 at 12:22