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(z_{1}) = f(z_{2}) implies z_{1} = z_{2} for z_{1}, z_{2} ∈ 𝔻. This tells you that a function f(z) defines a one to one correspondence 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*.

φ_{a} 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* + *b*i is *a* – *b*i (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:

**Regular starlike:**When mapping from the unit disk Δ to the complex plane ℂ, a function is starlike if and only if

(Nevanlinna, 1921).- 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

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