In complex analysis, the open mapping theorem is a domain of the complex plane is a non-constant holomorphic function is an open map, and we have invariance of domain.).

The open mapping theorem points to the sharp difference between holomorphy and real-differentiability. On the real line is not an open map, as the image of the open interval.

The theorem for example implies that a non-constant holomorphic function cannot map an open disk onto a portion of any line embedded in the complex plane. Images of holomorphic functions can be of real dimension zero (if constant) or two (if non-constant) but never of dimension 1.

Proof

is a domain of the complex plane. We have to show that every point is an interior point.

is a root of the function.

are isolated by the identity theorem is the only (although this single root may have multiplicity greater than 1).

is a circle and hence a compact set is a positive continuous function, so the extreme value theorem.

with radius. By Rouché's theorem.

is open.

Applications

• Maximum modulus principle

• Rouché's theorem

• Schwarz lemma

See also

• Open mapping theorem (functional analysis)

References