Univalent function

Examples

Any mapping ϕ a of the open unit disc to itself. The function

where | a | < 1 , 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 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

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 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).