schlicht functions


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 functionsMathworldPlanetmath. 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


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.


Let {fn} be a sequence of functions in S and fnf uniformly on compact subsets of the open unit disc. Then f is in 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 derivativeMathworldPlanetmath is 1 for functions in 𝒮 prevents this problem.


  • 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