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 is univalent in the open unit disc, as implies that. As the second factor is non-zero in the open unit disc, so is injective.

Basic properties

One can prove that if and are two open connected sets in the complex plane, and
is a univalent function such that , then the derivative of is never zero, is invertible, and its inverse is also holomorphic. More, one has by the chain rule
for all in

Comparison with real functions

For real analytic functions, unlike for complex analytic functions, these statements fail to hold. For example, consider the function
given by. 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 .