A Simple Proof of Bieberbach Conjecture

Abstract

The famous Riemann mapping theorem, which was established in 1851, states that every proper subdomain of the complex plane that is simply connected (without “holes”) is the image of the unit disk under a univalent analytic mapping f(z). This is the most significant characteristic of univalent analytic functions. The domain D, the image point f(0) in D, and the restriction that f’(0) be a positive real integer each independently define the mapping function f(z). One naturally asks how geometric aspects of the image domain f(D) represent analytic properties of f when considering univalent analytic functions as “Riemann mappings”. This is the perspective of geometric function theory, whose primary issue was solved by de Branges’ proof. The Bieberbach conjecture (hereinafter referred to as “BC”) is essentially a claim on the extreme nature of the Koebe function. Therefore, a key subject in this article is to make an explanation of the Koebe function and intepret why it is the best contender to be “largest” in the sense of the conjecture. Also, a detailed but easy understanding noval proof is given for this conjecture.