Consider *injective* homolomorphic functions $f:\mathbb D\to \mathbb C$ on the unit disk $|z|\leq 1$, normalized by the conditions $f(0)=0$ and $f'(0)=1$. 


Thus for $|z|\leq 1$ we have $ f(z)=\sum\_{k=0}^{\infty} a\_k z^k $ with $a\_0=0$ and $a\_1=1$.
Ludwig Bieberbach conjectured in 1916 and Louis de Branges proved in 1984 that for all $k \in \mathbb N$ the inequality $|a\_k|\leq k$ holds.
The Koebe function $K(z)=z/(1-z)^2$ does not belong to $ H^2(D) $ and $ A^2(D) $ . for which the extremal condition holds. Is there any analogous result for restricting the univalent functions to lie in $ H^2(D) $.


Although the above is settled there is another result, due to Hayman and independent of De Branges' theorem, that $ \lim a\_n/n \leq 1 $ and equality holds only for Koebe function. I think it might be worthwhile to find an analogue of this in $ A^2(D) $. In $ H^2(D) $, it doesn't make sense as $ \lim a\_n=0 $




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[$ A^2(D)$ is the bergman space with p=2](http://en.wikipedia.org/wiki/Bergman_space)



hayman's result for $ A^2(D) $
I am not sure what "analogous" means but the estimate $|a\_n|\leq n$ is best possible
when you restrict to $H^2$, and even when you restrict to polynomials. For the simple reason that the extremal function can be approximated by polynomials injective in the unit disc.


