Univalent Function
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Univalent Function

In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective.[1]

## Examples

show that the function

 f(z)=2z+z^2, |z|<1


Is univalent in the domain. Solve..

 Let z1 and z2 be any two points in |z|<1. then


f(z1)=f(z2)->(z1-z2)(z1+z2+2)=0 ->z1-z2=0 and z1+z2+2?0 in |z|<1 This is clear from the fact Re(z1+z2+2)=Rez1+Rez2+2>-1-1+2=0 Thus f is one-one in |z|<1 and hence univalent in the domain.

## Basic properties

One can prove that if ${\displaystyle G}$ and ${\displaystyle \Omega }$ are two open connected sets in the complex plane, and

${\displaystyle f:G\to \Omega }$

is a univalent function such that ${\displaystyle f(G)=\Omega }$ (that is, ${\displaystyle f}$ is surjective), then the derivative of ${\displaystyle f}$ is never zero, ${\displaystyle f}$ is invertible, and its inverse ${\displaystyle f^{-1}}$ is also holomorphic. More, one has by the chain rule

${\displaystyle (f^{-1})'(f(z))={\frac {1}{f'(z)}}}$

for all ${\displaystyle z}$ in ${\displaystyle G.}$

## Comparison with real functions

For real analytic functions, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold. For example, consider the function

${\displaystyle f:(-1,1)\to (-1,1)\,}$

given by ƒ(x) = x3. This function is clearly injective, but its derivative is 0 at x = 0, and its inverse is not analytic, or even differentiable, on the whole interval (−1, 1). Consequently, if we enlarge the domain to an open subset G of the complex plane, it must fail to be injective; and this is the case, since (for example) f(εω) = f(ε) (where ω is a primitive cube root of unity and ε is a positive real number smaller than the radius of G as a neighbourhood of 0).

## References

1. ^ John B. Conway (1996) Functions of One Complex Variable II, chapter 14: Conformal equivalence for simply connected regions, page 32, Springer-Verlag, New York, ISBN 0-387-94460-5. Definition 1.12: "A function on an open set is univalent if it is analytic and one-to-one."