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, : where $|a|le\; 1,$ is univalent.

Basic properties

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

:$f:\; G\; o\; Omega$

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

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

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

:$f:\; (-1,\; 1)\; o\; (-1,\; 1)$

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 0-387-90328-3.

* John B. Conway. "Functions of One Complex Variable II". Springer-Verlag, New York, 1996. ISBN 0-387-94460-5.

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