Options
Birth of the theory of univalent functions
Author
Lam, Hwee Peng
Supervisor
Ahuja, O. P.
Abstract
"Univalent" is the complex analyst's term for "one-to-one" functions. A univalent function has a non-vanishing derivative. Riemann Mapping Theorem stated in 1851 could not find some interesting applications until P. Koebe in 1907 discovered that univalent analytic functions have nice property as given in the modified version of Riemann Mapping Theorem. Koebe found some very exciting and useful results for the functions which are analytic and univalent in a simply connected domain. This gave birth to the Theory of Univalent Functions. Unfortunately, there is a long struggle for its growth and development.Only in 1914, the opportunity for this theory to grow was provided when Grownwall discovered the Area Theorem. This theorem is concerned with functions of the form
that are both analytic and univalent in the open disk
z+Σbnz-n
n=0
that are both analytic and univalent in the open disk
z
>1.
These functions are in close relation to the functions in a family S, which consists of analytic, univalent and normalized functions in the open unit disk I z I < 1. The family S plays a central role in the Theory of Univalent Functions. In 1916, Bieberbach discovered many interesting properties of the family S. This gave rise to much development in this theory. Finally, Bieberbach was encouraged to give his easy to state but difficult to prove conjecture concerning all the coefficients of the functions in the family S. The challenge to prove this conjecture had led many mathematicians to do research and hence, leading to much growth and development of this theory.
In this Academic Exercise, we shall look into the motivation for the birth of the Theory of Univalent Functions. We shall concentrate on properties found for the family S, with emphasis on the coefficient problems.
These functions are in close relation to the functions in a family S, which consists of analytic, univalent and normalized functions in the open unit disk I z I < 1. The family S plays a central role in the Theory of Univalent Functions. In 1916, Bieberbach discovered many interesting properties of the family S. This gave rise to much development in this theory. Finally, Bieberbach was encouraged to give his easy to state but difficult to prove conjecture concerning all the coefficients of the functions in the family S. The challenge to prove this conjecture had led many mathematicians to do research and hence, leading to much growth and development of this theory.
In this Academic Exercise, we shall look into the motivation for the birth of the Theory of Univalent Functions. We shall concentrate on properties found for the family S, with emphasis on the coefficient problems.
Date Issued
1998
Call Number
QA331 Lam
Date Submitted
1998