univalent analytic function


An analytic functionMathworldPlanetmath on an open set is called univalentMathworldPlanetmath if it is one-to-one.

For example mappings of the unit disc to itself ϕa:𝔻𝔻, where ϕa(z)=z-a1-a¯z, for any a𝔻 are univalent. The following summarizes some basic of univalent functions.


Suppose that G,ΩC are regions and f:GΩ is a univalent mapping such that f(G)=Ω (it is onto), then

  • f-1:ΩG (where f-1(f(z))=z) is an analytic function and (f-1)(f(z))=1f(z),

  • f(z)0 for all zG


  • 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
  • 2 John B. Conway. . Springer-Verlag, New York, New York, 1995.
Title univalent analytic function
Canonical name UnivalentAnalyticFunction
Date of creation 2013-03-22 14:12:06
Last modified on 2013-03-22 14:12:06
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 6
Author jirka (4157)
Entry type Definition
Classification msc 30C55
Synonym univalent function
Synonym univalent