Please use this identifier to cite or link to this item: http://hdl.handle.net/2445/151979
 Title: Univalent functions. The Bieberbach conjecture Author: Roig Sanchis, Anna Director/Tutor: Massaneda Clares, Francesc Xavier Keywords: Funcions univalentsTreballs de fi de grauFuncions de variables complexesTeoria geomètrica de funcionsUnivalent functionsBachelor's thesisFunctions of complex variablesGeometric function theory Issue Date: 19-Jun-2019 Abstract: [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. Note: Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2019, Director: Francesc Xavier Massaneda Clares URI: http://hdl.handle.net/2445/151979 Appears in Collections: Treballs Finals de Grau (TFG) - Matemàtiques

Files in This Item:
File Description SizeFormat