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′(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):= |
belongs to . So to study univalent functions on the unit disc it suffices to study . 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 be a sequence of functions in and uniformly on compact subsets of the open unit disc. Then is in .
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 prevents this problem.
References
- 1 John B. Conway. . Springer-Verlag, New York, New York, 1995.
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 |