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 one-to-one.

Examples

Any mapping $\phi _{a}$ of the open unit disc to itself,

$\phi _{a}(z)={\frac {z-a}{1-{\bar {a}}z}},$

where $||a||\leq 1,$ is univalent.

Basic properties

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

$f:G\to \Omega$

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

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

for all $z$ in $G.$

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

$f:(-1,1)\to (-1,1)$

given by $f(x)=x^{3}$. This function is clearly one-to-one, however, its derivative is 0 at $x=0$, and its inverse is not analytic, or even differentiable, on the whole interval $(-1,1).$

References

• John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, 1978. ISBN 0387903283.

• John B. Conway. Functions of One Complex Variable II. Springer-Verlag, New York, 1996. ISBN 0387944605.

univalent analytic function at PlanetMath.