Existence of univalent functions with prescribed coefficients. Let $f: \mathbb{D} \rightarrow \mathbb{C}$ be a univalent(injective) function with Taylor series $f(z)=z+a_2z + a_3 z^2 + \cdots $.
Here, $\mathbb{D}$ is the open unit disk.
It is well known that $|a_n| \leq n$ for all $n \geq 2$ (This is the so-called Bieberbach problem).
I want to ask if converse holds:
If the first two coefficients $|a_2| < 2$ and $|a_3| <3$ are prescribed, is there a function $f$ satisfying the above conditions?
If the answer to my original question is not known, then I want to know if there are any partial results to my question.
 A: Considering $\frac{1}{f(\frac{1}{z})}$  which is in $\Sigma$ (meromorphic functions $z+b_0+\sum_{n \ge 1}b_nz^{-n}$, analytic and univalent in the exterior of the unit disc, for which the area theorem gives $\sum_{n \ge 1}n|b_n|^2 \le 1$), hence for which $1 \ge |b_1|=|a_2^2-a_3|$, it follows that obviously even for the second and third coefficients there are clear restrictions. Fekete-Szego theorem gives more restrictions on such and then there are Hankel determinant restrictions etc. 
The only general theorem I know is that if $\sum_{n \ge 2}n|a_n| \le 1$, then $z+\sum_{n \ge 2}a_nz^n$ is starlike and the immediate analog for convex functions that follows from the Alexander criterion.
(Edit later - in particular if $2|a_2|+3|a_3| \le 1$ you can get a univalent function;  the book Coefficient Regions for Schlicht Functions, by Schaeffer & Spencer, AMS Colloquium Publications XXXV, 1950, has more or less the complete solution for the region $(a_2,a_3)$ at least its boundary which is composed of a few portions for which parametrizations with finitely many equations are given and some nice but quite weird pictures of the 3-D solids of such in the special cases $a_2$ real, or $a_3$ real when the region is of real dimension 3 of course; anyway, the solution is quite complicated even in this simple case of the first two non-trivial coefficients)
