# Topics in Complex Analysis

Share this page *Edited by *
*Dorothy B. Shaffer*

Most of the mathematical ideas presented in this volume are based on
papers given at an AMS meeting held at Fairfield University in October
1983. The unifying theme of the talks was Geometric Function Theory.

Papers in this volume generally represent extended versions of the
talks presented by the authors. In addition, the proceedings contain
several papers that could not be given in person. A few of the papers
have been expanded to include further research results obtained in the
time between the conference and submission of manuscripts. In most
cases, an expository section or history of recent research has been
added. The authors' new research results are incorporated into this
more general framework. The collection represents a survey of research
carried out in recent years in a variety of topics.

The paper by Y. J. Leung deals with the Loewner equation, classical
results on coefficient bodies and modern optimal control theory. Glenn
Schober writes about the class \(\Sigma\), its support points and
extremal configurations. Peter Duren deals with support points for the
class \(S\), Loewner chains and the process of truncation.

A very complete survey about the role of polynomials and their limits in
class \(S\) is contributed by T. J. Suffridge.

A generalization of the univalence criterion due to Nehari and its relation
to the hyperbolic metric is contained in the paper by David Minda. The omitted
area problem for functions in class \(S\) is solved in the paper by
Roger Barnard. New results on angular derivatives and domains are represented
in the paper by Burton Rodin and Stefan E. Warschawski, while estimates on the
radial growth of the derivative of univalent functions are given by Thom
MacGregor.

In the paper by B. Bshouty and W. Hengartner a conjecture of Bombieri is
proved for some cases. Other interesting problems for special subclasses are
solved by B. A. Case and J. R. Quine; M. O. Reade, H. Silverman and P. G.
Todorov; H. Silverman and E. M. Silvia.

New univalence criteria for integral transforms are given by Edward Merkes.
Potential theoretic results are represented in the paper by Jack Quine with
new results on the Star Function and by David Tepper with free boundary
problems in the flow around an obstacle. Approximation by functions which are
the solutions of more general elliptic equations are treated by A. Dufresnoy,
P.~M. Gauthier and W. H. Ow.

At the time of preparation of these manuscripts, nothing was known about
the proof of the Bieberbach conjecture. Many of the authors of this volume and
other experts in the field were recently interviewed by the editor regarding
the effect of the proof of the conjecture. Their ideas regarding future trends
in research in complex analysis are presented in the epilogue by Dorothy
Shaffer.

A graduate level course in complex analysis provides adequate background
for the enjoyment of this book.

# Table of Contents

## Topics in Complex Analysis

- Table of Contents v6 free
- Introduction vii8 free
- List of Speakers ix10 free
- Notes on Loewner Differential Equations 112 free
- Some Conjectures for the Class Σ 1324
- Truncation 2334
- Polynomials in Function Theory 3142
- The Schwarzian Derivative and Univalence Criteria 4354
- The Omitted Area Problem for Univalent Functions 5364
- Angular Derivative Conditions for Comb Domains 6172
- Radial Growth of a Univalent Function and its Derivatives off Sets of Measure Zero 6980
- Local Behaviour of Coefficients in Subclasses of S 7788
- Polygonal Bazilevič Functions 8596
- Coefficient Conditions for a Subclass of Alpha-Convex Functions 91102
- Classes of Rational Functions 99110
- The Quotient of a Univalent Function with its Partial Sum 105116
- Univalence of an Integral Transform 113124
- The Laplacian of the *- Function 121132
- A Jet Around an Obstacle 127138
- Runge's Theorem on Closed Sets for Elliptic Equations 133144
- Epilogue, The Bieberbach Conjecture 139150