An introduction to univalent function theory and the Bieberbach conjecture

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.