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 ${\displaystyle \phi _{a}}$  of the open unit disc to itself,  :${\displaystyle \phi _{a}(z)={\frac {z-a}{1-{\bar {a}}z}},}$  where ${\displaystyle |a|<1,}$  is univalent.

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 one-to-one; 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

• 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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