40 - Univalent Functions

Summary

Preliminary Remarks

Weierstrass constructed a theory of functions using power series as the basic object, in contrast with Riemann, who studied analytic functions as mappings, specifically conformal mappings. The Bieberbach conjecture was rooted in these dual aspects of analytic function theory; it simultaneously viewed a function as a mapping and as a series. Thus, Bieberbach considered conformal mappings, such as those studied by Riemann, and then speculated on the magnitude of the coefficients, assuming the first two to be zero and one, respectively. A function f analytic in a domain D, an open and connected subset of the complex plane, is called univalent in D if it does not assume any value more than once. A univalent function f maps D conformally onto its image domain f(D). Riemann was the first to study conformal mappings in the context of complex function theory. In his 1851 doctoral dissertation, he stated his famous theorem, now called the Riemann mapping theorem, that any simply connected proper subdomain D of the complex plane could be conformally mapped onto the unit disk |z| < 1. Note here that the mapping must be one-to-one and analytic. This mapping f is unique if we require that for a given point z₀ in the domain D, f(z₀) = 0 and f′(z₀) > 0. Observe that since the inverse of a univalent function is also univalent, it is of interest to consider functions univalent on the unit disk.