schlicht functions

Definition.

The class of univalent functions on the open unit disc in the complex plane such that for any $f$ in the class we have $f(0)=0$ and $f^{\prime }(0)=1$ is called the class of schlicht functions. Usually this class is denoted by $𝒮$.

Note that if $g$ is any univalent function on the unit disc, then the function $f$ defined by

$$f(z):=\frac{g(z)-g(0)}{g^{\prime }(0)}$$

belongs to $𝒮$. So to study univalent functions on the unit disc it suffices to study $𝒮$. A basic result on these gives that this set is in fact compact in the space of analytic functions on the unit disc.

Theorem.

Let $\mathrm{\{}{f}_n\mathrm{\}}$ be a sequence of functions in $\mathrm{S}$ and $f_n\mathrm{\to }f$ uniformly on compact subsets of the open unit disc. Then $f$ is in $\mathrm{S}$.

Alternatively this theorem can be stated for all univalent functions by the above remark, but there a sequence of univalent functions can converge either to a univalent function or to a constant. The requirement that the first derivative is 1 for functions in $𝒮$ prevents this problem.

Title	schlicht functions
Canonical name	SchlichtFunctions
Date of creation	2013-03-22 14:23:37
Last modified on	2013-03-22 14:23:37
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	8
Author	jirka (4157)
Entry type	Definition
Classification	msc 30C45
Synonym	schlicht function
Related topic	KoebeDistortionTheorem
Related topic	Koebe14Theorem