Abstract:
The Bieberbach Conjecture was formulated for the first time in 1916 by Ludwing Bieberbach and was studied in the following 68 years.
Let S denote the set of all univalent and analytic functions f defined in D = {z Î C : |z| < 1} such as f (0) = 0 e f' (0) = 1. Such functions have the expansion in Taylor series around zero

f  (z) = z + a2z2 +a3z3 +... ,|z| < 1

Bieberbach has conjectured the following:
The Bieberbach Conjecture: For each function f  Î S   we have |an| ² n, n = 1, 2, ...  . Furthermore, |an| = n para n = 1, 2, 3, ... only when f is the Koebe function or one of its rotations, i.e., f (z) = ß-1 k(ßz), with ß Î C, |ß| = 1 and k the Koebe function, k (z) = z + z2 + z3 + ... ,|z| < 1.
We will see that the Robertson Conjecture implies the Bieberbach Conjecture and the Milin Conjecture implies, on its turn, the Robertson Conjecture, being the problem reduced to the proof of the Milin Conjecture.
This dissertation is divided into four chapters. In the first one we intend to establish de.nitions as well as some basic results of Complex Analysis.
The second chapter presents a study on the Normal Families, considering the proof of the Riemann mapping theorem. In the third chapter, we will focus especi.cally on the Bieberbach Conjecture. This chapter includes the proof of the Bieberbach theorem (case n = 2), the proof of the Conjecture for starlike of S and for functions of S with real coficients and, finally, the De Branges proof of the Milin Conjecture. In appendix there is a brief study on the Möbius transformations used in the proof of the Riemann mapping theorem.
The results obtained on the univalent functions theory during the last decades are so vast and varied that it is for us impossible to show them here in its totality. This study presents only the most important ones, as well as the concepts stricty indispensable to the comprehension of the proof of the conjecture.