Suppose $f(z)=c\_1z+c\_2z^2+\cdots+c\_nz^n$ is a univalent map on the unit disk. 


You may assume $c\_1=1$. All coefficients are complex.


Is there a sharp bound on the modulus of the last coefficient $c\_n$? 


How about the other coefficients?



Bounds on coefficients: univalent maps
There is something to say about $c\_n$, if not much about the lower order coefficients.


Since $f(z)$ is a *schlicht* function, $f'(z)\neq0$ throughout the disk. That means, each root of the polynomial $f'(z)$ has modulus $\vert z\vert\geq1$. From
$$f'(z)=1+2c\_2z+\cdots+nc\_nz^{n-1},$$
we know that the product of the roots is $\frac1{nc\_n}$ and this quantity must satisfy
$$\frac1{n\vert c\_n\vert}\geq1.$$
In other words, $n\vert c\_n\vert\leq1$ or $\vert c\_n\vert\leq\frac1n$, which is indeed stronger than $\vert c\_n\vert\leq n$.


