 # Univalent Functions

Share on

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: 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 (Nevanlinna, 1921).
2. Close-to-convex: the definition is the same as regular starlike, except the function is convex if and only if:, ## 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!