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.

is univalent in the open unit disc, as

As the second factor is non-zero in the open unit disc,

are two open connected sets in the complex plane, and is a univalent function such that

is surjective), then the derivative of

is invertible, and its inverse

More, one has by the chain rule for all

For real analytic functions, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold.

This function is clearly injective, but its derivative is 0 at

, and its inverse is not analytic, or even differentiable, on the whole interval

Consequently, if we enlarge the domain to an open subset

of the complex plane, it must fail to be injective; and this is the case, since (for example)

f ( ε ω ) = f ( ε )

is a primitive cube root of unity and

is a positive real number smaller than the radius of

This article incorporates material from univalent analytic function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.