An invitation on the study of univalent and multivalent functions

A W Goodman

doi:10.1155/s016117127900017x

Abstract

We begin with the basic definition and some very simple examples from the theory of univalent functions. After a brief look at the literature, we survey the progress that has been made on certain problems in this field. The article ends with a few open questions.

Highlights

(i.i) in the complex variable z x + iy that are convergent in the unit disk zl E
Two questions present themselves: (A) given the sequence of coefficients b0, bl, b2, what can we say about the geometric nature of D: and (B) given some geometric property of D what can we say about the sequence b 0, b I, b 2, An example of a nice geometric property is given in DEFINITION i
-= As trivial examples, we mention that f(z) z is univalent in E while f2(z) zn/n z 2 is not univalent in E

Summary

Find a useful set of conditions on the sequence

{b n both necessary and sufficient for f(z) to be univalent in E The second volume covers the period from 1966 to 1975 and lists 1563 papers, Each of these ,2 volumes has an extenslve index which lists subtopics in this field and those papers that touch on each subtopic It is an easy matter for the specialist to determine the status of any problem up to the year 1975. Let PS(1) denote the set of all functions F(z) of the form (1.2) for which F(z) and all the partial sums are univalent in E. A function F(z) is said to be biunivalent in E if both F(z) and its inverse are univalent in E Let us examine this definition a little more closely.

Let r w be an arc of the boundary of

It is clear from the definitions of CV and ST that

As an example consider

Distler gave a beautiful generalization of these two theorems by finding

Izl and the Second Coefficient of a Univalent Function in