Timeline for Bieberbach-type bound for bounded univalent functions
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 12, 2016 at 21:30 | comment | added | Christian Remling | In the original paper, this is done slightly differently, by imposing a bound on $f$, but of course similar comments apply. | |
| Jun 12, 2016 at 21:26 | comment | added | Christian Remling | @AlexandreEremenko: Yes, I have read the result very carefully, though not in the original source, but in Garnett's book. In any event, the point of my comment was to point out the difference between an asymptotic bound, valid for $n\ge N_0(f)$, and a bound valid for all $f$ for a given $n$ (addressed by Lasse's answer also). | |
| Jun 12, 2016 at 21:25 | comment | added | Alexandre Eremenko | @Christian Remling: They do normalize their functions, and actually estimate $A_n$ which is the sup of $a_n$ over the normalized class. So the estimate is definitely better than $n$ for large $n$. | |
| Jun 11, 2016 at 16:00 | comment | added | Christian Remling | This is a very interesting result, but it's not clear if this answer the OP's question. C-J prove that $|a_n|\le Cn^{-b}$, with $b>0$ (note that they don't normalize by requiring $a_0=0$, $a_1=1$, this would be pointless in their setting); this just tells us that for any given $f$, eventually $|a_n|\ll n$, but it could still be that for a given $n$, all one can say in general is $|a_n|\le n$. | |
| Jun 11, 2016 at 14:49 | comment | added | Alan Sola | There are a couple of papers by Hedenmalm and Shimorin (Duke Math. J. '05, Proc AMS '07) on universal integral means spectra that may be relevant: as a byproduct of their results they get some further bounds for the decay rate of coefficients. | |
| Jun 11, 2016 at 8:04 | history | answered | Alexandre Eremenko | CC BY-SA 3.0 |