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 φ_a of the open unit disc to itself,

where is univalent.

Basic properties

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

is a univalent function such that f(G) = Ω (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

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

given by f(x) = x³. 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.