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
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 $$0$$).