Developments in the Theory of Univalent Functions

Abstract

A univalent function in a domain D is characterized by the property that it takes no value more than once in the domain and that, consequently, it maps the domain D onto a Schlicht domain, that is, one which is not self-overlapping and contains no branch points. In investigating the properties of analytic functions univalent in a domain D which is simply connected, one usually confines oneself to the case of the unit disc, for, by the Riemann mapping theorem, any simply connected domain with at least two boundary points can be mapped onto the unit disc, and any univalent function in D is associated with a corresponding univalent function in the unit disc. If f(z) is regular and univalent in the unit disc, so also is f(z) − f(0)}/f’(0), and this enables us to use the normalization f(0) = 0, f’(0) = 1 and study the class S of normalized regular functions having the Taylor expansion $$f(z) = z + {a_2}{z^2} + \; \cdot \cdot \cdot \;\;\;\;\;\left| z \right| < 1$$ We may observe that f’(0) does not vanish. If it did, f(z) could not be univalent. There also exist functions which are univalent but not regular in the unit disc. For example, if f(z) ∈ S, af(z) + b/ cf(z) + d, ad − bc ≠ 0 is obviously univalent in |z|<| but has a simple pole if −d/c is one of the values assumed by f(z) in |z| < 1.