An introduction to univalent function theory and the Bieberbach conjecture

TL;DR

This paper provides a comprehensive introduction to univalent function theory, detailing key theorems and presenting a full, self-contained proof of Bieberbach's conjecture for readers with basic analysis background.

Contribution

It offers a detailed, accessible exposition of univalent function theory and presents a complete proof of Bieberbach's conjecture, filling gaps left by previous texts.

Findings

Complete proof of Bieberbach's conjecture

Detailed exposition of univalent function theorems

Accessible introduction for learners

Abstract

The purpose of this paper is to make an introduction to univalent function theory for readers of any level, assuming only foundational knowledge in real and complex analysis. In particular, we state and proof (with details) important theorems utilised in proving Bieberbach's conjecture, especially those that were missed or merely sketched by other texts on univalent functions. We will finally prove the conjecture following the proof given by Lenard Weinstein in a comprehensive and self-contained manners, exposing its full technical details in each step. Readers are encouraged to make used of this paper to accompany the preliminary chapters of Duren's Univalent Functions and Lenard Weinstein's the Bieberbach Conjecture.