Univalent function

Examples

The function f \colon z \mapsto 2z + z^2 is univalent in the open unit disc, as f(z) = f(w) implies that f(z) - f(w) = (z-w)(z+w+2) = 0. As the second factor is non-zero in the open unit disc, z = w so f is injective.

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 f(x)=x^3. 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(\varepsilon \omega) = f(\varepsilon) (where \omega is a primitive cube root of unity and \varepsilon is a positive real number smaller than the radius of G as a neighbourhood of 0).

Note

References

is:Eintæk vörpun

References

1. {{harv. Conway. 1995
2. {{harv. Nehari. 1975