Calculus How To

Univalent Functions

Share on

Types of functions >

In general, a univalent function (sometimes called analytic univalent) is an analytic function that maps one input to one output (i.e. it is injective) on the complex plane. In other words, it’s a complex function that has no overlap.


More Narrow Definitions

Univalent Functions can also be defined more narrowly, as a function that maps different points to different points in the unit disk 𝔻 (Ravichandran, 2012).


In notation (Thomas, 1967), a function f(z) in a domain 𝔻 is univalent if w = f(z) takes on different values of w for different z ∈ 𝔻. In other words, f(z1) = f(z2) implies z1 = z2 for z1, z2 ∈ 𝔻. This tells you that a function f(z) defines a one to one correspondance between points of 𝔻 and a domain in the w-plane.

One example:
univalent function

Where:

  • z = a point in the complex plane,
  • a = the complex conjugate*.

φa is univalent when |a| < 1.

*A complex conjugate has a real part and an imaginary part with the same magnitude but different signs. For example, the complex conjugate of a + bi is abi (where a and b are two real numbers).

Some authors define univalent functions as being injective and meromorphic in the unit disk (e.g. Pommerenke, 1985). A meromorphic function is the ratio of two analytic functions, which are analytic with the exception of “poles“— isolated singularities.

Neighborhoods of univalent functions

An interesting feature of univalent functions is that their entire neighborhoods are also univalent functions (Pascu & Pascu, 2009). In other words, a small deviation in the function will not result in a non-univalent function.

Subclasses

Univalent functions can be categorized into various subclasses. For example:

  1. Regular starlike: When mapping from the unit disk Δ to the complex plane ℂ, a function is starlike if and only if
    starlike function
    (Nevanlinna, 1921).
  2. Close-to-convex: the definition is the same as regular starlike, except the function is convex if and only if:,
    convex

References

Nevanlinna, R. (1921). Über die konforme Abbildund Sterngebieten
Oversikt av Finska–Vetenskaps Societen Forhandlingar, 63(A) (6), pp. 48-403
Pascu,M. & Pascu, N. (2009). Neighborhoods of univalent functions. Retrieved December 19 from: https://arxiv.org/abs/0910.5456
Pemmerenke, Ch. (1985). On the Integral Means of the Derivative of a Univalent Function. Journal of the London Mathematical Society, Volume s2-32, Issue 2, October 1985, Pages 254–258, https://doi.org/10.1112/jlms/s2-32.2.254
Ravichandran, V. (2012). Geometric Properties of Partial Sums of Univalent Functions. Retrieved December 19, 2019 from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.746.6182&rep=rep1&type=pdf
Sokół, J. (2011). A certain class of starlike functions
Study, E. (1913). Konforme Abbildung Einfachzusammenhangender Bereiche
B. C. Teubner, Leipzig und Berlin
Thomas, D. (1967). Starlike and close-to-convex functions. Retrieved December 19, 2019 from: https://spiral.imperial.ac.uk/bitstream/10044/1/17592/2/Thomas-DK-1967-PhD-Thesis.pdf

CITE THIS AS:
Stephanie Glen. "Univalent Functions" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/univalent-functions/
------------------------------------------------------------------------------

Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!

Leave a Reply

Your email address will not be published. Required fields are marked *