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. 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. Consider the application ϕ a {displaystyle phi _{a}} mapping the open unit disc to itself such that We have that ϕ a {displaystyle phi _{a}} is univalent when | a | < 1 {displaystyle |a|<1} . One can prove that if G {displaystyle G} and Ω {displaystyle Omega } are two open connected sets in the complex plane, and is a univalent function such that f ( G ) = Ω {displaystyle f(G)=Omega } (that is, f {displaystyle f} is surjective), then the derivative of f {displaystyle f} is never zero, f {displaystyle f} is invertible, and its inverse f − 1 {displaystyle f^{-1}} is also holomorphic. More, one has by the chain rule for all z {displaystyle z} in G . {displaystyle G.} 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). This article incorporates material from univalent analytic function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.