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. == 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. 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 ).

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