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.

Examples

Any mapping $\phi_a$ of the open unit disc to itself. The function

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

where $|a|<1,$ 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 surjective), 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 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).

See also

• Schlicht function

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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