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

The function is univalent in the open unit disc, as f(z) = f(w) implies that. 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 is a univalent function such that (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 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) (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