Univalent Functions: A Primer

Univalent functions are analytic functions of a complex variable that are one-to-one on a given region (that is, they take no value more than once; hence the name). They used to go by the German name schlicht, even in English. They usually appear in a normalized form on the unit circle:

\( f(z) = z + a_2 z^2 + a_3 z^3 + \dots, \quad |z|< 1. \)

Univalent functions are interesting for several reasons, including: they appear in conformal mapping and in the Riemann mapping theorem; the inverse function exists and is also analytic; the set of univalent functions is closed in various senses (under composition and under uniform limits); and we can say some things about the coefficients. The Bieberbach conjecture (now a theorem of de Branges) states that \(|a_n| \le n\) (Bieberbach proved this for \(n = 2\)).

The present book carries the subtitle “A Primer”, and this is accurate (except that there are no exercises). It starts at the beginning. Up until the last three chapters it sticks to very elementary methods: typically these are ordinary calculation, inequalities, the maximum principle, Schwarz’s lemma, and the Rouché theorem. Univalent functions are divided into a number of classes, and the book primarily works on properties of the classes, with some introductory material that applies to all classes.

Although a primer, this is aimed at researchers, and there are a tremendous number of theorems. It also includes a final chapter on open problems, keyed to the applicable chapters. Most of the more-complete introductory complex analysis texts have sections on univalent functions, and if you just want to get started, these would be better choices. Boas & Boas’s Invitation to Complex Analysis has an especially good introduction in Section 35: just four pages, but it hits all the high points and has some exercises. Another well-regarded text (also for researchers, but also starting at the beginning), that I have not seen, is Duren’s Univalent Functions (Springer, 1983).

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.