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http://hdl.handle.net/2445/151979
Title: | Univalent functions. The Bieberbach conjecture |
Author: | Roig Sanchis, Anna |
Director/Tutor: | Massaneda Clares, Francesc Xavier |
Keywords: | Funcions univalents Treballs de fi de grau Funcions de variables complexes Teoria geomètrica de funcions Univalent functions Bachelor's theses Functions of complex variables Geometric 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 |
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File | Description | Size | Format | |
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151979.pdf | Memòria | 672.84 kB | Adobe PDF | View/Open |
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