Univalent functions. The Bieberbach conjecture

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2019, Director: Francesc Xavier Massaneda Clares[en] In this work, we will study the theory holomorphic and univalent functions in proper simply connected domains of $\mathbb{C}$; in particular on the case where the domain is the unit disk. We will expose the most important results of the theory, and focus especially on one of its major problems: the Bierberbach conjecture (BC), stated in 1916 by Ludwig Bieberbach, and proved in 1984 by Louis de Branges, which claims: Bieberbach's Conjecture. The coefficients of each analytic and univalent function $f(z)=$ $z+\sum_{n=2}^{\infty} a_{n} z^{n}$ in the unit disk, with $f(0)=0$ and $f^{\prime}(0)=1$ satisfy: $$ \left|a_{n}\right| \leq n, \quad \text { for } \quad n=2,3, \cdots $$ Strict inequality holds for every n unless $f$ is a rotation of the Koebe function